My background in electrical engineering and interest in teaching have led me to be most interested in applied mathematics and student projects. The following are some projects I have worked on or supervised.
Pattern recognition with Artificial Neural Networks
Artificial neural networks are a construction that mimics the large scale distributed processing capability of a biological brain. In graduate school, I worked with my advisor studying the effects of pre-transformations of input data on the resulting ability of the network to learn pattern classification.
25 years later, I oversaw the work of Adam Wroughton, an IWU undergraduate, as he successfully used a neural network to recognize musical chords.
Wroughton, Adam. Training and Testing an Artificial Neural Network for Musical Note Classification
Partial Differential Equations of Superconductivity
When certain materials get very cold, they lose essentially all resistance to the flow of electricity. However, a sufficiently strong external magnetic field can prevent this superconducting behavior from occurring. For my Ph.D. thesis, done under Patricia Bauman, I analyzed the Ginzburg-Landau partial differential equations to show they mathematically predicted the effects of a magnetic field that are actually observed in the laboratory.
Ginzburg-Landau Equations for a Three-Dimensional Superconductor in a Strong Magnetic Field, Ph.D. Thesis, Purdue University, 1997.
Two biological species which consume similar resources will compete with each other if placed in the same habitat. Numerous mathematical models, including discrete dynamical systems models, have been formulated to predict the two populations over time. Dr. Bob Mallison and I recently oversaw the work of Caryn Willis, and IWU undergraduate, who studied the effects of modifications of a model originally formulated by Hassel and Comins.
_Willis, Caryn. An Extension of the Hassel-Comins Discrete Time Model for Two Competing Species
Counter-intuitive Solids of Revolution
Calculus II students often encounter a solid of revolution know as Gabriel's Horn, which somewhat paradoxically turns out to have finite volume but infinite surface area. In other words, it is a solid that one can "fill but not paint." I compiled or generated several other solids (suitable for Calculus II or Real Analysis classes) with similar interesting geometrical properties (e.g. "fill but not slice" and "paint but not hike.")
Royer, Melvin. “Gabriel’s Other Possessions,” PRIMUS, vol 22, issue 4, 2012.
Parables to a Mathematician
Jesus frequently used parables in His ministry, usually short narratives illustrating the outcomes of people's choices. In John 3:12 and Matthew 13:10-15, He explained that one reason was to be sure that people who genuinely wanted to understand His message would be able to do so. Since most of His audience was familiar with an agrarian economy, Jesus spoke extensively of wheat, fish, trees, wine, debt, tenants, lamps, etc. Many people have speculated on parables Jesus might have used had He lived in a different society. This non-scholarly (but hopefully thought-provoking) talk proposed parables targeted toward groups of mathematicians with various levels of Christian background.
Royer, Melvin. Parables to a Mathematician, Conference Proceedings of the Association of Christians in the Mathematical Sciences, 2015
Axioms: Mathematical and Spiritual
Attempts in theology, psychology, and personal experience to find ultimate foundations (axioms) for beliefs have close parallels to the axiomatization of mathematics attempted by Euclid, Hilbert, Frege, Russell and Whitehead, and others. These mathematicians had varied success but learned much much about the nature, power, and limitations of deductive reasoning.
Considered as a parable, mathematical axiomatization and the use of models can help us understand the process of Christian apologetics and formulation of our own core spiritual beliefs. First, deductive reasoning in mathematics is a powerful illustration of the overall process of apologetics. There are differences, however, including the facts that spiritual reasoning involves more presuppositions and the axioms usually have a broader and more ambiguous scope. Next, working mathematicians often encounter difficulties finding consistent, independent, and complete axiom sets. Since formulating spiritual axioms is an even more open-ended process, it should not be surprising that a set of spiritual axioms will almost certainly be dependent and incomplete, and likely inconsistent as well. Finally, while the choices of both mathematical and spiritual axioms greatly impact the resulting system, there is evidence that a discipline can advance significantly even in the absence of a well-formed axiomatic base.
“Axioms: Mathematical and Spiritual, What Says the Parable,” Association of Christians in the
Mathematical Sciences Proceedings, Volume 21, 2018.
Digraphs of Commutative Rings
The successive iterations of a finite discrete dynamical system can be illustrated with a directed graph. Dr. Mallison and I oversaw the work of undergraduate student Morgan Bounds on the dynamical system mapping the ring Z(n) X Z(n) to itself under the operation (a,b) -> (a+b, ab) (mod n). Morgan was able to prove some open conjectures regarding the number of distinct cycles in the digraph. Bounds, Morgan "New Theorems for the Digraphs of Commutative Rings"