Discrete Modeling of Biological Systems
Professor Alan Veliz Cuba (Department of Mathematics) and Eric Eager (Graduate Student, Department of Mathematics)
The REU students studied how the design of experiments affects the network inference problem, which is to use the data of a dynamical system to infer the network. The goal was to determine how the generation of the initial data affects the inferred network.
We compared two ways of generating data:
- by running the system as long as possible for few initializations (few long timecourses);
- running the system for short periods of time, but many initializations (many short timecourses).
We did this for different published models of gene regulatory networks.
The question is, given a fixed sample size, is it better to have few long timecourses or many short timecourses? If the sample size is small, then generating data using few long timecourses will produce a better inferred network. However, when the sample size increases, there is a point where generating data using many short timecourse is better. This is an interesting result, because it was not expected for the sample size to matter so much. The students presented a poster at the 2013 JMM.
Professor Allan Peterson (Department of Mathematics) and Tanner Auch (Graduate Student, Department of Mathematics)
The REU students extended many results known in the q-calculus (quantum calculus, which has very important applications in quantum theory). This includes a q-Laplace transform, which has several similar properties to the continuous Laplace transform.
They have a large number of results on the properties of their q-Laplace transform. They have a clever definition of exponential growth of a function and with their definition are able to show regions of convergence of the q-Laplace transform in the numerous formulas that they derive. They use their q-Laplace transform to show how they can solve the so called $q$-difference equations. Finally, they use their q-Taylor monomials to obtain a Variation of Constants Formula.
The students presented a poster at the 2013 JMM.
Peridynamics Models in Heat Conduction and Elasticity
The students worked in the area of peridynamics, a theory introduced by S. Silling that has been successful in modeling phenomena in materials with discontinuities, or propagation of fractures. We establish connections between classical local operators and their nonlocal counterparts, and identify convergence rates for these models.
The students also derived a nonlinear diffusion model in the nonlocal framework of peridynamics, following ideas of Bobaru and Duangpanya. For the case when the conductivity is time dependent a fundamental solution for the nonlocal problem was derived, and they we proved an exponential decay rates by using energy methods and a nonlocal version of the Poincare’s inequality.
Both papers present numerical simulations that illustrate estimates for the solution in the quasilinear case as well as in the case of time dependent conductivity.