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American Mathematical Monthly -May 2008

May 2008

For subscribers, read The American Mathematical Monthly online.

Sudoku, Gerechte Designs, Resolutions, Affine Space, Spreads, Reguli, and Hamming Codes
By: R. A. Bailey, Peter J. Cameron, and Robert Connelly
r.a.bailey@qmul.ac.uk, p.j.cameron@qmul.ac.uk, rc46@cornell.edu
Instructions for solving Sudoku puzzles often say "There's no math involved; you solve the puzzle with reasoning and logic." We refute this by showing how Sudoku is connected with the mathematical and statistical topics in the title. The solution to a Sudoku is a special case of a "gerechte design," a type of design introduced for agricultural experiments; we discuss features the statistician considers in choosing a design. The cells of the Sudoku grid can be coordinatized by a 4-dimensional vector space over the 3-element field; we show how all solutions satisfying some extra conditions can be found, using ideas from finite projective and affine geometry and from coding theory.

 

When is a Periodic Function the Curvature of a Closed Plane Curve?
By: J. Arroyo, O. J. Garay, and J. J. Mencía
mtparolj@ehu.es, oscarj.garay@ehu.es, mtpmegoj@ehu.es
The curvature of a closed plane curve is a periodic function, but the converse of this statement is not true in general and gives rise to a classical problem in differential geometry: The Closed Curve Problem. So the point is: what additional constraints must a periodic function satisfy in order to assure the closedness of its associate plane curve?

We show that if the signed area bounded by a periodic function in its minimum period is a non-integer rational multiple of 2π, then its associate plane curve closes up. We offer both a geometric and an analytical proof of this fact. As a consequence, any plane curve arising from a periodic function can be forced to close up, just by increasing or decreasing by the same amount its bending at each point. Finally, we discuss the case not covered by our condition along with other related questions.

 

Rearing Its Ugly Head: The Cosmological Constant and Newton's Greatest Blunder
By: Hieu D. Nguyen
nguyen@rowan.edu
It is folklore that Albert Einstein’s greatest blunder occurred when he introduced the cosmological constant force into his theory of general relativity to account for what was then believed to be a static universe. Not so well known is how this cosmological force first reared its ‘ugly’ head with Isaac Newton, who made a similar blunder in the Principia by introducing an extraneous force into his lunar theory to explain the advance of the moon’s apsis. In this article, we expose how both forces are actually one and the same from the perspective of Newtonian mechanics. Along the way we reveal Newton’s beautiful theory of revolving orbits, developed as a result of his blunder, and apply it to calculate (correctly this time) the precession of Mercury’s perihelion, one of the hallmark tests of general relativity.

 

The Riemann Hypothesis for Elliptic Curves
By: Jasbir S. Chahal and Brian Osserman
jasbir@math.byu.edu, osserman@math.berkeley.edu
The Riemann hypothesis for elliptic curves over finite fields, conjectured by E. Artin and proved by H. Hasse, is analogous to the classical Riemann hypothesis. However, where the classical version is equivalent to a statement on the distribution of prime numbers, the case of elliptic curves is equivalent to a uniform bound on the numbers of points on the curves. We explain in elementary terms how both Riemann hypotheses are special cases of a single more general conjecture, and we give an elementary proof of the Riemann hypothesis for elliptic curves in terms of counting points.

 

Notes

Inflating the Cube Without Stretching
By: Igor Pak

Yet Another Proof of Minkowski's Inequality
By: Robert Kantrowitz and Michael M. Neumann
rkantrow@hamilton.edu, neumann@math.msstate.edu

Extreme Palindromes
By: Kathy Q. Ji and Herbert S. Wilf
ji@nankai.edu.cn, wilf@math.upenn.edu

Thin Sets With Fat Shadows: Projections of Cantor Sets
By: F. Mendivil and T. D. Taylor
franklin.mendivil@acadiau.ca, ttaylor@stfx.ca

The n-Dimensional Pythagorean Theorem via the Divergence Theorem
By: Larry Eifler and Noah H. Rhee
eiflerl@umkc.edu rheen@umkc.edu

Derivatives and Eulerian Numbers
By: Grzegorz RzÄ…dkowski
g.rzadkowski@uksw.edu.pl

Reviews

Euler Through Time: A New Look at Old Themes.
By: V. S. Varadarajan
Reviewed by: Ranjan Roy
royr@beloit.edu

King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry.
By: Siobhan Roberts
Reviewed by: Mark E. Kidwell
mek@usna.edu

 

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