|August 6, 2012|
The Mathematical Association of America has selected P. Mark Kayll (University of Montana) and John A. Adam (Old Dominion University) as the 2012 winners of the Carl B. Allendoerfer Award. Full citations and biographical information for each winner are available below.
The Carl B. Allendoerfer Awards, established in 1976, are given for articles of expository excellence published in Mathematics Magazine. The Awards are named for Carl B. Allendoerfer, a distinguished mathematician at the University of Washington and President of the Mathematical Association of America, 1959-60. This is an award of $500. Up to two of these awards are given annually at the Summer Meeting of the Association.
?Integrals Don't Have Anything to Do with Discrete Math, Do They?? Mathematics Magazine, 84:2 (2011), p. 108-119.
Mathematical work is often so highly specialized that mathematicians in one field can find it difficult to understand research in other areas. These divisions are frequently reflected in mathematics education where courses such as ?Discrete Mathematics? suggest the compartmentalization of discrete and continuous topics. However, there are many examples of contemporary problems that combine disparate mathematical fields’such as algebra, geometry, topology, and combinatorics’or that connect mathematics to seemingly unrelated disciplines, such as biology. It is in this context that Mark Kayll enthusiastically reminds us that integrals do have something to do with discrete mathematics.
The beauty of Kayll's article lies in its exposition of some not-so-well known integral formulas for the number of perfect matchings in a graph. Deftly alternating between discrete and continuous topics, the author expresses the number of perfect matchings in a complete bipartite graph in terms of the gamma function. After this initial combination of the discrete and continuous, he expands his collection of improper integrals with the introduction of rook polynomials and derangements. This development of topics continues to a final full refutation of the author's title, a proof that the number of perfect matchings in a complete graph on n vertices is the nth moment of a standard normal random variable.
Kayll's well-written article provides us with engaging examples in which discrete and continuous mathematics come together. It reminds us that elements such as the gamma function are interesting in their own right, and it elegantly illustrates some of the ways in which continuous mathematics can be used to study discrete concepts. Enough details are included for the reader to follow the story, and comprehensive references are provided for those who want to learn more. Readers will finish the article with an increased appreciation of how surprising connections can exist between the discrete and the continuous, and of how the teaching of these subjects as distinct entities can be misleading to students.
Mark Kayll grew up in North Vancouver, British Columbia. After earning mathematics degrees from Simon Fraser University (B.Sc. 1987) and Rutgers University (Ph.D. 1994), he joined the faculty at the University of Montana in Missoula. He's enjoyed sabbaticals in Slovenia (University of Ljubljana, 2001?02) and Canada (UniversitÃ© de MontrÃ©al, 2008?09).
His publications fall in the discrete realm and have touched on combinatorics, graph theory, number theory, and probability. Mark's musical interests, such as playing the banjo, have motivated him in recent years to develop a general education course on mathematics and music for non-math majors.
He lives in Missoula with his wife, Jennifer (an excellent editor), and two beautiful children, Samuel and Leah.
?Blood Vessel Branching: Beyond the Standard Calculus Problem,? Mathematics Magazine, 84:3 (2011), p. 196-207.
What optimality principles determine the structure of the arteries, veins, and capillaries that comprise the human circulatory system? How reasonable are estimates that the total length of all the blood vessels within the body is on the order of 50,000 miles? This insightful and intellectually rich article offers an approach to these questions by studying the flow, branching, and maintenance of this important biological tree. To carry out his analysis, the author considers a sequence of increasingly sophisticated models. The first of these models employs only standard calculus tools to determine the optimal branching from a straight blood vessel to a nearby point. Later models use techniques from the calculus of variations to optimize a configuration for a combination of flow and volume.
The author first lays the groundwork by discussing the underlying biological setting, specifying his simplifying assumptions, and introducing some necessary equations from fluid dynamics. He then develops a sequence of models for blood vessel branching based upon a series of ever more comprehensive ?cost functionals.? The most sophisticated of these models implies certain empirical laws for vascular branching proposed by Wilhelm Roux in 1878, and also yields estimates for the total length of the vascular system.
This well-written article provides an excellent example of mathematical modeling in a context that is accessible and of obvious importance. It clearly shows the interaction of appropriate mathematical techniques with relevant scientific principles and illustrates the complexity of the modeling process. The reader is left with a deeper understanding of the power of mathematics to shed light on natural phenomena.
John Adam is Professor of Mathematics at Old Dominion University. He received his Ph.D. in theoretical astrophysics from the University of London in 1975. He is author of approximately 100 papers in several areas of applied mathematics and mathematical modeling. His first book, Mathematics in Nature: Modeling Patterns in the Natural World, was published in 2003 by Princeton University Press (paperback in 2006). He enjoys spending time with his family, especially his (thus far) five grandchildren, walking, nature photography, and is a frequent contributor to the Earth Science Picture of the Day (EPOD: http://epod.usra.edu/).
In 2007 he was a recipient of the State Council of Higher Education of Virginia's Outstanding Faculty Award. He co- authored Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, published by Princeton University Press in 2008. More recently he has authored A Mathematical Nature Walk (2009, paperback version in 2011) and X and the City: Modeling Aspects of Urban Life (2012), both published by Princeton.