Research

Research


Research Areas:

Nonlinear Conservation Laws: The idea that certain quantities in systems are conserved is a fundamental idea in science and are an avenue for writing down equations describing how these systems evolve.  The system of Euler questions, modeling nonlinear gas dynamics, is one of these systems.  One particular consequence of these equations is that waves of different density and pressure propagate at different velocities.  These different velocities cause the solution to “fold” on it self in finite time producing a discontinuous solution in finite time (even from smooth initial conditions.)  Much work in the field of Nonlinear conservation laws is done trying to analyze the dynamics of these discontinuities (shock waves) and elementary waves, including these moving discontinuities, interact.

Time Series Analysis: When we observe phenomena only a subset of the underlying quantities that govern the system may be visible.  With this limited knowledge, can we reconstruct some of the overall (hidden) dynamics of a system?  This is one of the goals of Time Series Analysis.  Characteristics such as dimension and Lyapunov exponent are estimated, as well as looking for the “best” reconstruction the systems’ attractor.  While difficult to justify, this analysis may suggest whether or not a system exhibits chaos.

Factorization Theory in Monoids: We are familiar that each natural number can be uniquely factored into products of prime numbers.  It turns out that that natural numbers are a Monoid, a set of elements with an associative binary operation (multiplication) that has an identity element, 1.  We can consider other Monoids generated by the natural numbers such as the McNugget monoid, the set of numbers generated by adding any combinations of 6, 9 and 20’s.  In this monoid, where the binary operation is plus, elements may no longer have unique factorizations into irreducible elements.  (In contrast to factorization into the primes)  Various questions can be studied about factorization properties of elements in these (numerical) monoids. 

Publications:

Rebecca Conaway, Felix Gotti, Jesse Horton, Christopher O'Neill, Roberto Pelayo, Mesa Williams, Brian Wissman, "Minimal presentations of shifted numerical monoids", Pre-print, https://arxiv.org/abs/1701.08555

Katherine F. Benson, Daniela Ferrero, Mary Flagg, Veronika Furst, Leslie Hogben, Violeta Vasilevskak, Brian Wissman, "Power domination and zero forcing",  Pre-print, http://arxiv.org/abs/1510.02421

B. D. Wissman, L. C. McKay-Jones, and P.-M. Binder, "Entropy rate estimates from mutual information", Physical Review E 84 (2011): 046204

P.-M Binder and B.D. Wissman, "Geometry of Repeated Measurements in Chaotic Systems". CHAOS 20, 013106 (2010).

B.D. Wissman, “Global solutions to the ultra-relativistic Euler equations”. Comm. in Math. Phys., 306(3) (2011), 831-851.