Changing Time History in Moving Boundary Problems

A class of diffusion-stress equations modeling transport of solvent in glassy polymers is considered. The problem is formulated as a one-phase Stefan problem. It is shown that the moving front changes like √t initially but quickly behaves like t as t increases. The behavior is typical of stress-dominated transport. The quasi-steady state approximation is used to analyze the time history of the moving front. This analysis is motivated by the small time solution

Bayesian source separation of fMRI signals

In analyzing the results of functional magnetic resonance imaging, the identification of significant activation in voxels is a crucial task. In computing the activation level, a standard method is to select an assumed to be known reference function and perform a multiple regression of the time courses on it and a linear trend. Once the linear trend is found, the correlation between the assumed to be known reference function and the detrended observed time-course in each voxel is computed and voxels colored according to their correlation. But the most important question is: How do we choose the reference function? This paper develops a Bayesian statistical approach to determining the underlying source reference function based on Bayesian source separation, and uses it on both simulated and real fMRI data. This underlying reference function is the unobserved response due the presentation of the experimental stimulus

An Unusual Moving Boundary Condition Arising in Anomalous Diffusion Problems

In the context of analyzing a new model for nonlinear diffusion in polymers, an unusual condition appears at the moving interface between the glassy and rubbery phases of the polymer. This condition, which arises from the inclusion of a viscoelastic memory term in our equations, has received very little attention in the mathematical literature. Due to the unusual form of the moving-boundary condition, further study is needed as to the existence and uniqueness of solutions satisfying such a condition. The moving boundary condition which results is not solvable by similarity solutions, but can be solved by integral equation techniques. A solution process is outlined to illustrate the unusual nature of the condition; the profiles which result are characteristic of a dissolving polymer

Shock Formation in a Multidimensional Viscoelastic Diffusive System

We examine a model for non-Fickian "sorption overshoot" behavior in diffusive polymer-penetrant systems. The equations of motion proposed by Cohen and White [SIAM J. Appl. Math., 51 (1991), pp. 472–483] are solved for two-dimensional problems using matched asymptotic expansions. The phenomenon of shock formation predicted by the model is examined and contrasted with similar behavior in classical reaction-diffusion systems. Mass uptake curves produced by the model are examined and shown to compare favorably with experimental observations

Fast spatial inference in the homogeneous Ising model

The Ising model is important in statistical modeling and inference in many applications, however its normalizing constant, mean number of active vertices and mean spin interaction are intractable. We provide accurate approximations that make it possible to calculate these quantities numerically. Simulation studies indicate good performance when compared to Markov Chain Monte Carlo methods and at a tiny fraction of the time. The methodology is also used to perform Bayesian inference in a functional Magnetic Resonance Imaging activation detection experiment.Comment: 18 pages, 1 figure, 3 table