As part of the Natural Science and Mathematics Colloquium, American University mathematics professor Dan Kalman, father of St. Mary’s graduate Chris Kalman ’05, presented Province of Polynomia – Uncommon Excursions for the Seasoned Visitor on Feb. 28.
On March 9, Susquehanna University professor and St. Mary’s graduate Jonathan Niles brought the audience back to the United States with a discussion of trout-stream ecology in his lecture Riparian Forest and Stream Interactions: The Importance of Terrestrial Invertebrates to Brook Trout in Appalachian Streams.
Introduced by mathematics professor Alex Meadows, Kalman began his presentation with a discussion of roots and polynomials, intended for a mixed audience of students, faculty, and community members with varied understandings of Kalman’s field.
Creating a mathematical world known as “Polynomia” as a means of organizing his lecture, Kalman explained how polynomials have roots, or powers of one that satisfy the polynomial if they replace the variable “x”, usually when the polynomial is set to equal zero.
He showed the audience how to use such knowledge by solving any polynomial p(x), where p(x) was a polynomial with any degree of non-negative integer coefficients, simply by knowing what the polynomial would equal if “x” was replaced by one and what it would be if that solution also replaced “x”.
For those not math-savvy in the audience, Kalman showed that if an equation p(x) was modified so that “x” was replaced by “1/x”, the numbers in front of those variables would all switch in sequence once the equation was set up as a fraction, with the denominator being the highest-powered “x” variable.
“It’s not as applicable as a talk about trout,” said Kalman, foreshadowing the following week’s lecture with Niles, but nonetheless seemed to be an effective application of the concept.
Continuing with a discussion of curly roots to solve cubic polynomials, Newtonian identities, synthetic division to divide equations, binary powering as a means of representing functions, and Lill’s Method as an older, graphical method of solving a polynomial’s roots, Kalman concluded with an important point for all math and non-math venturers of Polynomia.
“There are many more Polynomia destinations,” he said. “I hope you come again soon.”
The following week, Niles continued the series with his discussion of a very different area of research: how the plant and animal life on the banks of streams have a major influence on the ecology of aquatic organisms. Introduced by biology professor Bob Paul as a former student of Paul’s Limnology class, Niles began with a discussion of what is a riparian forest.
“Most people think that ‘riparian’ refers to the long stretch [of land] on the side of the stream … but this is not the case,” said Niles, “It does run parallel, but also is dependent on the land up the slopes. It is bidirectional.”
After previous research showed that the energy in streams on the west coast of the United States was not sufficient to support the large fish population, and that riparian organisms landing in the stream for fish consumption were taking up the energy requirements, Niles tested this idea with brook trout populations in Appalachian streams.
He prepared eight streams, four with 50 percent of the canopy around the streams cut and four with 90 percent of the canopy cut, with a uncut regions to serve as a control, to observe the change in diets of the brook trout over an almost one-month period. He did so by observing internal stomach contents via gastric lavage (flooding of the body with water to induce regurgitation).
Niles determined there was a higher amount of terrestrial biomass in trout from the uncut regions compared to the 90 percent cut regions, indicating the importance of riparian life on trout diets.
“Niles showed an interesting relationship between terrestrial invertebrates and the ecosystem,” said biology and biochemistry major Luke Trout, junior, “but there seemed to be a lot of variables that would make me question the validity of the experiment.”
The next NS&M lecture will be on March 23 by SUNY-Potsdam mathematics professor Joel Foisy.