March 2012 Contents
In March’s American Mathematical Monthly, we honor John Ewing, the winner of the 2012 Gung and Hu Award for Distinguished Service to Mathematics. Dr. Ewing is a former Editor of the Monthly, and Ann Watkins offers us a summary of his career. Our articles for March give us an introduction to Algebraic Geometry, a lesson in stereographic projections, and sufficient conditions for the existence of median Mitscherlich and Verhulst curves. Notes include viewing square roots as homomorphisms, a new approach to Cauchy random variables, and character analysis using Fourier Series. Gerald Alexanderson reviews Hersh and John-Steiner’s Loving and Hating Mathematics: Challenging the Myths of Mathematical Life. Last but not least, don’t forget our Problem Section.
Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2012 to John Ewing for Distinguished Service to MathematicsAnn E. Watkins
About the Gung-Hu Award. The Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics is intended to be the most prestigious award for service made by the Association. It honors distinguished contributions to mathematics and mathematical education—in one particular aspect or many, and in a short period or over a career. The initial endowment was contributed by husband and wife Dr. Charles Y. Hu and Yueh-Gin Gung, a professor of geography at the University of Maryland and a librarian at the University of Chicago, respectively. They contributed generously to our discipline because, as they wrote, “We always have high regard and great respect for the intellectual agility and high quality of mind of mathematicians and consider mathematics as the most vital field of study in the technological age we are living in.”
An Introduction to Algebraic Geometry: Polygons, Parameterizations, and EquationsJessica Sidman
We explain how four surfaces that are ubiquitous in algebraic geometry may be obtained from convex polygons whose vertices have integer coordinates, realizing these surfaces as examples of toric varieties. We highlight the combinatorial and algebraic properties revealed by this construction. Our claim of novelty is in our particular combination of the toric, classical geometric, and algebraic points of view.
The Mercator and Stereographic Projections, and Many in BetweenDaniel Daners
We consider a family of conformal (angle preserving) projections of the sphere onto the plane. The family is referred to as the Lambert conic conformal projections. Special cases include the Mercator map and the stereographic projection. The techniques only involve elementary calculus and trigonometry.
Real and Generic Data without Unconstrained Best-Fitting Verhulst Curves and Sufficient Conditions for Median Mitscherlich and Verhulst Curves to ExistYves Nievergelt
A commonly taught scientific method for building mathematical models uses finite computations to approximate the curve of a specified type that best fits the data, without checking whether any such best-fitting curve exists: not every regression objective need have a global unconstrained minimum. One counterexample will confute its theoretical foundation: any triple of points with super-exponential growth does not admit of any unconstrained bestfitting Verhulst logistic curve, regardless of the regression criterion. Moreover, because the set of all such triples is open, there are still no best-fitting Verhulst curves after sufficiently small but arbitrary perturbations of the data. Nevertheless, the present explanations show that through each triple of points with sub-exponential growth passes a unique Verhulst curve. Furthermore, if every triple of data grows sub-exponentially, then for the reciprocal data there exists a median Mitscherlich curve whose reciprocal is a Verhulst curve. Applications range from alchemy to zoology.
Square Root as a Homomorphism
William C. Waterhouse
In any field K where −1 is not a square, there are homomorphisms from the squares to (chosen) square roots in K. Maximal fields F with such homomorphisms have either or the 2-adic integers.
Sample Means of Independent Standard Cauchy Random Variables Are Standard Cauchy: A New ApproachMichael P. Cohen
A remarkable property of the Cauchy distribution is that the sample mean of standard Cauchy random variables itself has a standard Cauchy distribution. This result can be shown using characteristic functions, convolution integrals, or multidimensional change of variables (Jacobians). A new method of proof based on the periodicity properties of the tangent function is presented.
Character Analysis Using Fourier SeriesDaniel Reem
We show that character analysis using Fourier series is possible, at least when a mathematical character is considered. Previous approaches to character analysis are somewhat not in the spirit of harmonic analysis.
PROBLEMS AND SOLUTIONS
REVIEWSLoving and Hating Mathematics: Challenging the Myths of Mathematical Life by Reuben Hersh and Vera John-Steiner (Princeton University Press)
Reviewed by Gerald L. Alexanderson