Research

Highlighted Publications

Der-Chen Chang (with Ovidiu Calin and Peter Greiner) Geometric Analysis on the Heisenberg Group and Its Generalizations, AMS/IP Studies in Advanced Mathematics, 2007, Volume: 40
Abstract: The theory of subRiemannian manifolds is closely related to Hamiltonian mechanics. In this book, the authors examine the properties and applications of subRiemannian manifolds that automatically satisfy the Heisenberg principle, which may be useful in quantum mechanics. In particular, the behavior of geodesics in this setting plays an important role in finding heat kernels and propagators for Schrödinger's equation. One of the novelties of this book is the introduction of techniques from complex Hamiltonian mechanics.

Sellers, K.F., (with Miecznikowski, J.C., Viswanathan, S., Eddy, W.F., and Minden, J.) (2007) "Lights, Camera, Action: Quantitative Analysis of Systematic Variation in Two-dimensional Difference Gel Electrophoresis", Electrophoresis, 28 (18), 3324-3332.
Abstract: 2-D Difference gel electrophoresis (DIGE) circumvents many of the problems associated with gel comparison via the traditional 2-DE approach. DIGE's accuracy and precision, however, is compromised by the existence of other significant sources of systematic variation, including that caused by the apparatus used for imaging proteins (location of the camera and lighting units, background material, imperfections within that material, etc.). Through a series of experiments, we estimate some of these factors, and account for their effect on the DIGE experimental data, thus providing improved estimates of the true relative protein intensities. The model presented here includes 2-DE images as a special case.


Paul C. Kainen, Vera Kurkova, and Andrew Vogt. "A Sobolev-type upper bound for rates of approximation by linear combinations of Heaviside plane waves." J. of Approximation Theory 147.1 (2007): 1-10 .
Description: We show that a linear combination of n Heaviside plane waves can approximate the Gaussian function in d dimensions with a mean-square error bounded above by (2 pi d)^(3/4) n^(-1/2). This means that radial-basis units can replace Heaviside units in neural networks with only a mild increase in the size of the network. The work is based on a more general theorem regarding functions which are sufficiently rapidly decreasing at infinity. Detemining the formula for the Gaussian involved an estimate of the total variation of derivatives of the Gaussian - i.e., the total amount of work one would need to do to climb up and down all the hills surrounding the central peak.


J.R. Miller (with B.P. Wood), Linked selected and neutral loci in heterogeneous environments, J. Math. Biol.53 (2006), pp. 939-975. 
 Abstract: We analyze a system of ordinary differential equations modeling haplotype frequencies at a physically linked pair of loci, one selected and one neutral, in a population consisting of two demes with divergent selection regimes. The system is singularly perturbed, with the migration rate m between the demes serving as a small parameter. We use geometric singular perturbation theory to show that when m is sufficiently small, each solution not initially fixed for the same selected allele in both demes approaches one of a 1-dimensional continuum of equilibria. We then obtain asymptotic expansions of the solutions and show their validity on arbitrarily long finite time intervals. From these expansions we obtain formulas for the transient dynamics of FST (a measure of population structure) at both loci, as well as for the rate of genotyping error if the allelic state at the selected locus is inferred from that at the neutral (marker) locus. We examine two cases in detail, one modeling two populations in secondary contact after a period of evolution in allopatry, and the other modeling the origination and spread of a resistance allele.

T. Luo (with J. Smoller) Rotating fluids with self-gravitation in bounded domains, Arch. Rational Mech. & Anal. (2004)
Abstract: In this paper, we study the steady solutions of Euler-Poisson equations in bounded domains with prescribed angular velocity. This models a rotating Newtonian star consisting of a compressible perfect fluid with given equation of state. We obtain both existence and non-existence theorems, depending on the adiabatic gas constant. In addition, for the spherically symmetric solutions, which is a good approximation of rotating stars with slow rotation, we obtain some interesting properties of the solutions; e.g., monotonicity of the radius of the star with both angular velocity and central density. We also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is physically striking and in sharp contrast to the case of the non-rotating star. For general domains and variable angular velocities, both an existence result for the isentropic equations of state and non-existence result for the non-isentropic equation of state are also obtained.

James Sandefur, Elementary Mathematical Modeling A Dynamic Approach, Thomson Learning (2002) Description: ELEMENTARY MATHEMATICAL MODELING uses mathematics to study problems arising in areas such as Genetics, Finance, Medicine, and Economics. Throughout the course of the book, students learn how to model a real situation, such as testing levels of lead in children or environmental cleanup. They then learn how to analyze that model in relationship to the real world, such as making recommendations for minimum treatment time for children exposed to lead paint or determining the minimum time required to adequately clean up a polluted lake. Often the results will be counterintuitive, such as finding that an increase in the rate of wild-life harvesting may actually decrease the long-term harvest, or that a lottery prize that is paid out over a number of years is worth far less than its advertised value. This use of mathematics illustrates and models real-world issues and questions, bringing the value of mathematics to life for students, enabling them to see, perhaps for the first time, the utility of mathematics.

Chronological List of Selected Publications