American Mathematical Monthly -December 2008

December 2008

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The Quest for a Fuzzy Tychonoff Theorem
By: Stephan C. Carlson
carlson@rose-hulman.edu
In 1965 L. A. Zadeh introduced the notion of fuzzy sets, which generalize subsets of a given set by assigning to each point of the set a numerical membership value between zero and one, and this simple idea inspired research in the field of fuzzy logic as well as in areas of pure mathematics. One of the earliest such areas considered was fuzzy topology, in which one goal was determining the extent to which the notions of point set topology are preserved when subsets of a set are replaced by fuzzy sets, and this area is still an active field of research today. This article offers a retrospective tour of some results in fuzzy topology obtained between Zadeh's 1965 introductory work and R. Lowen's 1977 proof of a version of the Tychonoff theorem that asserts the compactness of a product of arbitrarily many compact fuzzy topological spaces. Along the way lie some interesting surprises involving such ideas as lower semicontinuity, lattice theory, cardinal numbers, category theory, and moreÂ—all contributing to a great story of how ideas develop in the process of mathematical research.

Surprising Sinc Sums and Integrals
By: Robert Baillie, David Borwein, and Jonathan M. Borwein
rjbaillie@frii.com, dborwein@uwo.ca, jborwein@cs.dal.ca
We intend to show that a variety of trigonometric sums have unexpected closed forms by relating them to cognate integrals. We hope this offers a good advertisement for the possibilities of experimental mathematics, as well as providing both some entertaining examples for the classroom and a caution against over-extrapolating from seemingly compelling initial patterns.

The Maximum Average Gain in a Sequence of Bernoulli Games
wolfgang@mathematik.uni-osnabrueck.de
The author considers a sequence of games where one flips a (possibly biased) coin and wins $1 if it comes up heads and loses$1 if it comes up tails. At any time, one can compute the average gain for the games that have been played so far. The maximum average gain that will be achieved takes only rational values, and each rational number in a certain interval is attained with positive probability. In this paper the exact distribution of this random variable is derived in closed form.

Developing Assessment Methodologies for Quantitative Literacy: A Formative Study
By: Jack Bookman, Susan L. Ganter, and Rick Morgan
bookman@math.duke.edu, sganter@clemson.edu, rlmorga@uncg.edu
Quantitative Literacy (QL) is a relatively new and unexplored area in higher educationÂ—one in which there is growing interest and whose importance, especially in today's economy, cannot be overstated. Studies that inform and support this rapid growth must be a higher priority in educational research, in order to establish QL as a viable and measurable area of student learning. This paper will address some of the issues related to these critical (though largely unexamined) questions about QL. In this paper, we discuss initial efforts to assess QL and methods for developing appropriate assessment instruments, and, in particular, tests used for student placement and proficiency levels. The following will be addressed: (1) item analyses for proficiency tests with partial credit scoring; (2) item analyses for proficiency tests with 0/1 scoring; (3) implications for the development of QL assessments; and (4) preliminary results using pre- and post-tests to evaluate the effectiveness of classroom instruction in quantitative literacy.

Notes

Another Continued Fraction for π
By: Thomas J. Pickett and Ann Coleman
tpickett@usi.edu

A Curious Result Related to Kempner's Series
By: Bakir Farhi
bakir.farhi@gmail.com

The Summation of a Family of Series
By: Andrei Vernescu
avernescu@clicknet.ro

A Devil's Platform
By: Washek F. Pfeffer
washek@mcn.org

Reviews

Information and Complexity in Statistical Modeling.
By: Jorma Rissanen
Reviewed by: Ioannis Kontoyiannis
yiannis@aueb.gr