Awards for exemplary writing in MAA journals and magazines were handed out in August at MAA MathFest in Hartford, Connecticut. The winning articles (PDF) are now available online, along with the award citations explaining why each article merited an award.
Carl B. Allendoerfer Awards (made to authors of articles of expository excellence published in Mathematics Magazine)
Khristo N. Boyadzhiev
Close Encounters with the Stirling Numbers of the Second Kind
Mathematics Magazine, Vol. 85, no. 4, October 2012, pp. 252-266
Citation: The Scottish mathematician James Stirling, in his 1730 book Methodus Differentialis, explored Newton series, which are expansions of functions in terms of difference polynomials. The coefficients of these polynomials, computed using finite differences, are the Stirling numbers of the second kind. Curiously, they arise in many other ways, ranging from scalar products of vectors of integer powers with vectors of binomial coefficients to polynomials that can be used to compute the derivatives of tan x and sec x.
This well-written exploration of Stirling numbers visits the work of Stirling, Newton, Grünert, Euler, and Jacob Bernoulli. Boyadzhiev’s fascinating historical survey centers on the representation of Stirling numbers of the second kind by a binomial transform formula. This might suggest a combinatorial approach to the study, but the article is novel in its analytical approach that mixes combinatorics and analysis. Grounded in Stirling’s early work on Newton series, this analytical approach illustrates the value of considering alternatives to Taylor’s series when expressing a function as a polynomial series.
The story of Stirling numbers continues with the exponential polynomials of Johann Grünert and geometric polynomials in the works of Euler. Boyadzhiev shows the relation of Stirling numbers of the second kind to the Bernoulli numbers and Euler polynomials. The article closes with a brief look at Stirling numbers of the first kind, a nice touch that deftly brings the proceedings to a close.
Boyadzhiev’s lively exposition engages the reader and leaves one eager to learn more.
Adrian Rice and Ezra Brown
Why Ellipses Are Not Elliptic Curves
Mathematics Magazine, Vol. 85, no. 3, June 2012, pp. 163-176
Citation: While ellipses and elliptic curves are two topics most mathematicians know something about, few of us have considered how they relate to each other. It is clear that the equations of ellipses and elliptic curves are different, but why then are their names so similar? This excellent exposition explores where the related names came from despite the core differences in these two famous mathematical objects.
The authors begin with a brief history of ellipses, starting in Ancient Greece. We then quickly arrive at elliptic integrals, which first arose from the desire to compute the arc length of a section of an ellipse. This historical tour ends with Jacobi and Eisenstein working with the doubly periodic properties of elliptic functions, the inverses of elliptic integrals. This engaging article returns to the Greeks, with the focus on elliptic curves this time, then progresses to Fermat, goes on to Newton, and finally returns to Eisenstein to connect elliptic curves to elliptic functions. It is here that the two subjects come together; they are both connected, in their own way, to elliptic functions.
The authors then give us the final twist: having parameterized elliptic curves using elliptic functions and ellipses using trigonometric functions, they show us that the parameterization of an ellipse in complex space is topologically a sphere, whereas the parameterization of an elliptic curve in complex space is topologically a torus.
Rice and Brown’s well-written article weaves the story of these two diverse mathematical objects, giving key historical and mathematical references along the way. Their engaging tour of mathematical history illustrates both how these two objects are related and why, mathematically, they are fundamentally different.