April 2010
“Despite the increasing importance of mathematics to
the progress of our economy and society, enrollment
in mathematics programs has been declining at all
levels of the American educational system. Yet the
application of mathematics is indispensable in such
diverse fields as medicine, computer sciences, space
exploration, the skilled trades, business, defense, and
government. To help encourage the study and
utilization of mathematics, it is appropriate that all
Americans be reminded of the importance of this
basic branch of science to our daily lives.”
In recent years, ecologists have begun to refine the
mathematical study of complex networks of
interactions between species within an ecosystem. In
particular, Network Environs Analysis (NEA) has
made use of linear algebra to formalize analysis of the
relative importance of direct and indirect connections
between species. One of the many indicators used to
quantify such description is homogenization, which is
a measure of the propensity of a network to distribute
material throughout all species along all
connections. We examine a matrix-theoretic
formulation of the homogenization indicator and
describe how linear algebraic tools can be refined to
determine which species and paths may exert strongest
influence on the behavior of an ecosystem.
Ecologists and conservation biologists often need to know
what roles species perform in ecosystems in order to
make informed management decisions and allocate
limited resources most effectively. However,
quantifying the functional roles of species has proved
difficult because species are embedded in complex
networks of interactions and a species’ importance
cannot be understood in isolation. Here we introduce
environ centrality (EC), a metric that quantifies the
relative roles of species in generating ecosystem
activity and demonstrate its functionality over four
empirically-based ecosystem network models. Our
results highlight the relative evenness of species roles
in ecosystems and demonstrates how indirect
relationships promote this evenness.
Mathematical modeling of the spreading of air pollutants in a windy atmosphere
has become a crucial field of study over the past several decades. With the
advances in computer technology and development of sophisticated
programming languages, scientists have become able to evaluate evolutionary
partial differential equations using numerical methods with a high degree of
accuracy. Computer simulations of air pollution dynamics will help to reveal
crucial properties about the forces interacting in gaseous systems, and will
also help scientists to create accurate models that predict global weather
patterns, climate change, global warming, air pollution spread through cities,
and many other critical studies. Two very common numerical methods for
solving evolutionary partial differential equations will be presented in this
research; finite difference methods and finite volume methods. A thorough
investigation into the accuracies and faults between each of these two
drastically different methods will be discussed. A detailed discussion on the
hydrodynamical equations used to model the spread of air pollution will be
approached in an elementary, but intense physical sense. The parameters
involved, as well as the source terms that provide the forcing for the emission
of pollutants, will be introduced and thoroughly dissected. An analysis of the
mass density profiles of the transport of pollutants in the atmosphere of the
Earth obtained from this study will be presented at the end. A discussion
about the accuracy of the theoretical simulations and its relationship between
actual experimental data will ultimately be considered.

Student Awards and Pi Mu Epsilon Ceremony