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The Graph Menagerie: Abstract Algebra and the Mad Veterinarian

Year of Award: 2011

Publication Information: Mathematics Magazine, vol. 83, 2010, pp. 168-179

Summary: This article begins with a fanciful concept from recreational mathematics: a machine that can transmogrify a single animal of a given species into a finite nonempty collection of animals from any number of species. Given this premise, a natural question arises: if a Mad Veterinarian has a finite slate of such machines, then which animal menageries are equivalent? To answer this question, the authors associate to the slate of machines a directed "Mad Vet" graph. They then show that the corresponding collection of equivalence classes of animal menageries forms a semigroup and use the structure of the Mad Vet graph to determine when this collection is actually a group. In addition, the authors show that the Mad Vet groups can be identified explicitly using the Smith normal form of a matrix closely related to the incidence matrix of the Mad Vet graph.

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About the Authors (From the MathFest 2011 Prizes and Awards Booklet)

Gene Abrams is Professor of Mathematics at the University of Colorado Colorado Springs. He earned his Ph.D. in mathematics at the University of Oregon in 1981. He has been actively involved in mathematics-oriented community outreach K-12 educational activities. Gene has published research articles and lectured extensively (both in the U.S. and Europe) on topics in associative rings and their modules, focusing since 2005 on Leavitt path algebras, and has been honored with various teaching awards, including: 2002 Teacher of the Year (Mathematical Association of America Rocky Mountain Section), President‟s Teaching Scholar (University of Colorado system, a lifetime designation made in 1996), and 1988 Outstanding Teaching Award (UCCS campus).

When he's not riding his bicycle, Gene succumbs to his passions for baseball, skiing, and the New York Times Sunday Crossword. He has been married to Mickey since 1983; they have two children, Ben and Ellen.

Jessica K. Sklar is Associate Professor of Mathematics at Pacific Lutheran University in Tacoma, Washington. She received her Ph.D. in mathematics at the University of Oregon in 2001. She is enamored of recreational mathematics and previously published an article in Mathematics Magazine (79:5, 2006) on the use of linear algebra in solving computer game puzzles. She is passionate about teaching and about sharing the beauty of mathematics with lay readers; she won Pacific Lutheran University's Faculty Excellence in Teaching Award in 2005–2006. Jessica is currently co-editing a collection of essays on mathematics in popular culture with Elizabeth Sklar. She lives in Seattle with her three charming and impish cats.

MSC Codes: 
05-XX
Author(s): 
Gene Abrams (University of Colorado Colorado Springs) and Jessica K. Sklar (Pacific Lutheran University)
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Publication Date: 
Sunday, August 21, 2011
Publish Page: 
Summary: 
This article begins with a fanciful concept from recreational mathematics: a machine that can transmogrify a single animal of a given species into a finite nonempty collection of animals from any number of species. Given this premise, a natural question arises: if a Mad Veterinarian has a finite slate of such machines, then which animal menageries are equivalent? To answer this question, the authors associate to the slate of machines a directed "Mad Vet" graph. They then show that the corresponding collection of equivalence classes of animal menageries forms a semigroup and use the structure of the Mad Vet graph to determine when this collection is actually a group. In addition, the authors show that the Mad Vet groups can be identified explicitly using the Smith normal form of a matrix closely related to the incidence matrix of the Mad Vet graph.

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