Vanderbilt Mathematics
Analysis & Biomathematics Seminar
Fall 2001

    Monday, December 3rd.
Wai-Yuan Tan, of the Department of Mathematical Sciences, University of Memphis. Some State Space Models and Generalized State Space Models in AIDS and Cancer. State space models and generalized state space models consist of many submodels linked together by multi-level Gibbs sampling method. Hence one may view state space models as a method to combine information from different sources. In this talk, I will illustrate the basic idea of state space models and generalized state space models and demonstrate its applications to some AIDS and cancer problems. A generalized Bayesian method is introduced to illustrate how to use these models to estimate simultaneously the unknown parametwers and state variables. This will combine information from three different sources: The stochastic system model of the system ( mechanism ), the statistical information from data generated from the system and prior information about the unknown parameters. The method is illustarted using the chain binomial model of HIV epidemic in homosexual population. Also, I will illustrate how to link stochastic models of molecular events such as DNA adduct and cell signal transduction of carcinogenesis at the molecular level to critical events at the cellular and population level in carcinogenesis in animal initiation and promotion experiments.

    Wednesday, November 28th.
Jason Moore, of the Program in Human Genetics, Vanderbilt University. New Paradigms for the Analysis of High-Dimensional Genetic Data. One goal of human genetics is to identify genes that confer an increased risk of common chronic diseases such as cardiovascular diseases. New technologies are making is possible to measure tremendous amounts of information about human genes. Despite these technological advances, the analytical tools to make sense of the vast amounts of data being generated have not kept pace. An overview of the analytical challenges that we face in the identification of disease susceptibility genes will be presented. The recently developed mutifactor dimensionality reduction (MDR) method will be presented as an alternative to traditional parametric statistical methods such as logistic regression for modeling high-dimensional genetic data.

    Thursday, November 15th, 3:10p.m. DOUBLE FEATURE
Eugene Eckstein, of the Department of Biomedical Engineering, University of Memphis & The University of Tennessee, Memphis. Measurements in Flowing Suspensions and Mathematical Models. Flow-associated mixing of cells in suspensions, including blood, is much stronger than would be expected from simple Brownian motion. This outcome is often attributed to unknown, but crucial initial information (e.g., a spatial distribution for particle positions and momenta). Choosing the appropriate mathematical models to describe such flows and events is a continuing challenge. Workers currently use "apparent parameters" (e.g., effective diffusion coefficient and effective viscosity) in classical equations of transport phenomena (Navier-Stokes and convective diffusion equations). Interesting medical applications involving mixing and suspension flows occur for sizes and conditions at which the continuum limit for using the equations is not obviously correct. Examples include the delivery to, adhesion on, or injury of cells at surfaces from streams with radii of a few to hundreds of cell diameters.
    In concentrated suspension flows, the use of apparent parameters is particularly troublesome because both the viscosity and diffusivity strongly depend on the concentration in an unknown way. The Stokes-Einstein equation, which links viscosity and diffusivity, is little direct use. Yet Einstein’s viewpoint and the stochastic description of particle motions are basic parts of collecting data. The mathematical description of Brownian motion of spherical particles in a fluid (Ornstein-Uhlenbeck process) provides an algebraic formula relating the mean squared displacement and the observation time, which reduces to the Stokes-Einstein equation in the limit of long time. The same formula is found for a persistent random walk model, discussed for turbulence by GI Taylor, and linked to the telegraph equation by S. Goldstein and M. Kac. The algebraic formula is logically linked to both parabolic and hyperbolic forms of partial differential equations. Our data fit this equation.
    Our experimental method provides a means of measuring the persistence of labeled cell motion down the average pressure gradient associated with a steady suspension flow. Observations have been made both for dilute concentrations of small, labeled particles and for small fractions of labeled particles among concentrated blood suspensions. Each observation was collected in a reference frame that moves at the apparent initial condition (of axial speed or equally the axial rate of change of mechanical energy). And for each initial condition, a nested set of net changes with respect to the initial condition was measured. The character of the axial dispersion of the labeled particles was demonstrable in ensembles of such data because they fit very well (p < 0.001) to the algebraic formula. Interestingly, the fitted parameter that is traditionally called a friction coefficient has a negative sign.
    This work was a collaborative effort with Ma Baoshun, Jerome Goldstein, Mohammad Kiani, JoDe Lavine and Marko Leggas.

    Thursday, November 15th, 4:10p.m. DOUBLE FEATURE
Jerome Goldstein, of the Department of Mathematical Sciences, University of Memphis. Mathematical Aspects of Suspensions. The random walk models of 1905-1922 lead to both parabolic (heat) and hyperbolic (telegraph) equations to describe certain diffusive phenomena. In trying to explain certain blood flow experiments made in Gene Eckstein's laboratory, it was discovered that the telegraph equation fits the data better than the heat equation. Moreover the Fürth-Ornstein-Taylor formula (1917-1922) fits it better than either of the above equations. We shall discuss various aspects of this circle of ideas, including fractional derivative telegraph equations (motivated by self-similarity considerations) and a heat equation with a delay term, derived from the F-O-T formula. The main punch lines are that solutions of the telegraph equation are asymptotically like solutions of the heat equation ("asymptotic analyticity of a group of operators") and solutions of the delay - heat equation are asymptotically like solutions of a generalized telegraph equation. This is joint work with Gene Eckstein and other teammates.

    Wednesday, November 7th.
Robert D. Tanner, of the Department of Chemical Engineering, Vanderbilt University. Foam Fractionation of Proteins. In an effort to develop low cost, simple and scaleup-able processes to separate and concentrate proteins we are developing a system to convert the desired proteins into a foam by adding a gas to the system. Once the hydrophobic protein foam phase is formed it can be skimmed off the top. An overview of the field will be presented starting with the example of a glass of beer. Other examples to be presented include the formation of foams of egg albumin and cellulase. A simple staged countercurrent model of the process will be briefly presented to illustrate the difficulty of mathematically describing this separation technique.

    Wednesday, October 31st.
Boris Kupershmidt, of the Department of Mathematics, University of Tennessee Space Institute. The Worst Thing That's Ever Happened to the Burger's Equation. It's its exact linearization via the Hopf-Cole transformation. Most of the subtle properties of the Burger's Equation and the corresponding commuting hierarchy are not related to the linearization, and their discovery was thus delayed for many decades. Similar but more complex conclusions apply to most known integrable systems, such as the KdV equation, many conjecturally so far. The new subject could be called "Dark Equations, Invisible Flows, Hidden Integrals."

    Wednesday, October 24th.
Ken Stephenson, of the Department of Mathematics, University of Tennessee, Knoxville. Mathematics of Brain Flattening: The Case for "Circle Packing". Patterns of functional brain activity in brain scan data are difficult to analyse because of the highly convoluted nature of the human cortex. Cortical "flattening" is intended to address these problems by mapping the cortical sheet to a flat surface. In this talk I will outline the math/computation issues in brain flattening, with special emphasis on a new "conformal" approach. Conformal maps of such complicated surfaces have become possible only with the advent of "circle packing" methods; there is a chance that some deep mathematics can play a pivotal role in this burgeoning neuroscience topic. I will describe methods, show examples, pose open issues. There are potentially many other uses for our novel mapping tools, both in math and science, so I look forward to exchanges with the audience. (Collaborators: Chuck Collins (UTK), De Witt Sumners, Phil Bowers, and Monica Hurdal (FSU), and David Rottenberg (Minn).

    Wednesday, October 17th.
Jeffrey Schall, of the Department of Psychology, the Center for Integrative and Cognitive Neuroscience and the Vision Research Center, Vanderbilt University. Neural Signals for Selecting Targets and Controlling Movements. We monitor the activity of neurons while subjects direct gaze to visual stimuli. Some neurons contribute to selecting the target for a movement of the eyes. Other neurons contribute to producing the movement of the eyes. The purpose of this talk is to introduce the kind of data we collect, the analyses we perform and the open questions. For example, it is well known that large populations of neurons are necessary to carry out these functions. However, paradoxically, we have found that the signals produced by very few neurons are sufficient to account for the performance. How can these two facts be reconciled?

    Monday, October 8th.
Bo Su, of the Department of Mathematics, University of Wisconsin, Madison. Discontinuous Solutions of Hamilton-Jacobi Equations: Existence, Uniqueness, and Regularity. In this talk, we address the discontinuous solutions of Hamilton-Jacobi equations. The existence of $L^{\infty}$ solutions is proven for general Hamiltonians. Then we clarify the connections in between the existing notions including the classical semicontinuous viscosity solution by Ishii. We look at the important special class of Hamiltonians and show the uniqueness of discontinuous solutions including the classical semicontinuous viscosity solutions, $L^{\infty}$ solutions. We prove the new interesting regularity property for locally strictly convex Hamiltonians $H$ such as $H(p)=(1+|p|^2)^{\frac 12}$. In other words, with discontinuous initial data, the discontinuous solutions of $u_t+H(Du)=0$ becomes Lipchitz continuous in finite time. The difference between the regularization effect of $u_t+|Du|^2=0$ and $u_t+(1+|Du|^2)^{\frac12}=0$ is similar to the difference between the difference of regularity of $u_t-\Delta u=0$ and $u_t-\Delta u^2=0$.

    Wednesday, September 26th.
Shigui Ruan, of the Department of Mathematics and Statistics, Dalhousie University and the Department of Mathematics, Vanderbilt University. On a Diffusive Population Equation with Distributed Delay. In this talk we will first review some well-known results on single species models such as the Multhusian model, logistic model, Hutchinson's model, Nicholson's blowflies model, Volterra integrodifferential equation model, etc. Then we consider the generalized Nicholson's blowflies model described by a scalar diffusive integradifferential equation. Stability of the steady state solutions, Hopf bifurcation, and existence traveling front solutions will be discussed.