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American Mathematical Monthly -March 2009

March 2009

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2009 to Robert E. Megginson for Distinguished Service to Mathematics
By: Barbara Faires
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Working with Schools: The Story of a Mathematics Education Collaboration
By: Alan H. Schoenfeld
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Working for meaningful mathematical change in the schools isn't easy. There are issues of politics, turf, and sometimes unreasonable expectations on the part of the school district and the volunteers who work with it. But with good intentions, goodwill, and tenacity, there are ways to make a difference. This paper describes some of the ups, the downs, and the ultimate progress in a collaboration between U.C. Berkeley and the Berkeley Unified School district. It offers lessons to mathematicians who want to understand and/or work with their local schools.


Some Integrals Involving the Cantor Function
By: Russell A. Gordon
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Since the Cantor function is developed in graduate-level real analysis as a counterexample for some properties of the derivative, very few people consider what happens if this function is treated like the more familiar continuous functions that appear in elementary integral calculus. In particular, what can be said about $\int_0^1f\circ c$, where f is a continuous function? It turns out that the value of this integral is closely related to the midpoint rule approximations of $\int_0^1f$. Some interesting formulas for the exact values of these integrals can be found when fis a power function, an exponential function, or a sine or cosine function.


Disentangling a Triangle
By: Jerzy Kocik and Andrzej Solecki
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When Ptolemy inscribed a triangle into a unit-diameter circle, its sides became visualizations of ratios that were later named "sines". His construction is known as the "sine theorem". We complete this idea and propose a visualization of the complementary ratios in this triangle, namely cosines, and a dual "cosine theorem".

A whole sequel of theorems on the geometry of triangles follows in a natural and easy way. The trigonometric sine-cosine duality finds a nice resolution with the help of certain quadruples, called unit orthotetrads. And there is no need anymore for trigonometric identities to be wrapped up inside an algebraic nightmare—just look and see.


Overshooting Properties of Newton-Like and Ostrowski-Like Methods
By: Aaron Melman
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Consider a situation where Newton's method is used to compute the largest zero of a polynomial with all real zeros, starting from a point to the right of that zero. The convergence will be monotonic from the right. If one doubles the Newton step to accelerate the convergence, progress towards the zero will still be monotonic, but only as long as the iterates do not overshoot the zero. How far can the iterates overshoot? A classical result shows that an overshoot will always land to the right of the largest zero of the derivative of the polynomial. What we will show here is that Newton's method is not unique in this respect and that similar properties hold for other methods, starting with Ostrowski's square-root method. We then show that the properties of Newton's and Ostrowski's methods are just particular manifestations of a more general property of a larger class of methods to which both belong.


Dual Billiards, Fagnano Orbits, and Regular Polygons
By: Serge Troubetzkoy
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Consider a polygon Q. Connect the midpoints of the sides to form a new polygon P. This procedure is closely related to the game of dual billiards: if Q intersects P only at vertices of P then Q is a dual billiard orbit for the table P. In this article we study the dynamical system QP and use it to characterize regular and affinely regular polygons.



The Poincaré Lemma and an Elementary Construction of Vector Potentials
By: Shirley Llamado Yap
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The Poincaré lemma guarantees that every conservative force field F defined on Rn is the differential of a potential field. In this paper, we provide an elementary construction of such potential fields.

Seating Couples Around the King's Table and a New Characterization of Prime Numbers
By: Emmanuel Preissmann and Maurice Mischler
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A king invites n couples to sit around a round table with 2n + 1 seats. For each couple, we decide a prescribed distance d between 1 and n which the two spouses have to be seated from each other (distance d means that they are separated by exactly d - 1 chairs). Is it true that there always exists a solution for every possibility if and only if 2n + 1 is a prime number? We give an affirmative answer to this problem using an unusual polynomial.

Superabundant Numbers and the Riemann Hypothesis
By: Amir Akbary and Zachary Friggstad
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Robin's criterion for the Riemann Hypothesis states that the Riemann Hypothesis is true if and only if $$\frac{\sigma(n)}{e^{\gamma}~ n \log{\log{n}}}


Mr. Hopkins' Men: Cambridge Reform and British Mathematics in the 19th Century.
By: Alex D. D. Craik
Reviewed by: Philip D. Straffin
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