A "Unified Theory" for Calculus

A University of Missouri-Rolla mathematician's research into a "unified theory" of continuous and discrete calculus is gaining the attention of mathematicians worldwide for numerous applications, including the study of insect populations.

Dr. Martin Bohner, an assistant professor of mathematics and statistics at UMR, introduced his work on dynamic equations and time scales at the American Mathematical Society's annual meeting in January 2002 in San Diego. Bohner's paper, "Asymptotic Behavior of Dynamic Equations on Time Scales," was recently cited as "Fast Breaking" by ISI Essential Science Indicators. The designation means the work represents scientific contributions that are just beginning to attract the attention of the scientific community. Bohner co-authored the paper with Dr. Donald Lutz of San Diego State University.

ISI Essential Science Indicators lists highly cited papers in 22 broad fields of science which comprise the top one percent of papers in each field.

Bohner's paper had the highest percentage increase in citations in ISI Essential Science Indicators in the field of mathematics from the second to third bimonthly periods of 2002.

"This paper is part of a fairly new and exciting effort to unify continuous and discrete calculus," says Bohner. "Dynamic equations on time scales have been introduced in order to unify the theories of differential equations and of difference equations and in order to extend those theories to other kinds of so-called 'dynamic equations.'"

The study of time scales has led to such applications as the study of insect population models. "Time scales calculus has a tremendous potential for applications," Bohner says. "It can model insect populations that are continuous while in season, die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population."

Further applications of time scales include neural networks, heat transfer and epidemic models. "Many researchers are getting interested in this theory and are contributing important results," Bohner says.

At UMR, Bohner teaches a variety of courses in the mathematics and statistics department, including matrix algebra, calculus, differential equations and engineering statistics.

For more information see http://esi-topics.com/fbp/fbp-october2002.html or http://web.umr.edu/~bohner.