Christopher P. Grant

SPINODAL DECOMPOSITION FOR THE CAHN-HILLIARD EQUATION

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. When a spatially homogeneous alloy is rapidly quenched in a physical experiment, a fine-grained decomposition into two distinct phases is frequently observed; this phenomenon is known as spinodal decomposition. A simple linear analysis about an unstable homogeneous equilibrium of the one-dimensional Cahn-Hilliard equation gives heuristic evidence that most solutions that start with initial data near such an equilibrium exhibit a behavior corresponding to spinodal decomposition. In this paper we formulate this conjecture in a mathematically precise way, using geometric and measure-theoretic techniques, and prove its validity. We believe that this is the first rigorous treatment of this phenomenon.

An extension of Zermelo's model for ranking by paired comparisons

In 1929, Zermelo proposed a probabilistic model for ranking by paired comparisons and showed that this model produces a unique ranking of the objects under consideration when the outcome matrix is irreducible. When the matrix is reducible, the model may yield only a partial ordering of the objects. In this paper, we analyse a natural extension of Zermelo’s model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use. Suppose that n objects are compared a pair at a time, and that for each comparison one of the two objects in the pair is judged superior to the other. (A common example would be athletic teams engaged in pairwise competitions.) The results of the comparisons can be summarized in the outcome matrix A =( aij), where aij is the number of comparisons in which object i is judged to be superior to object j. If all of the o-diagonal elements in A +A T are the same (i.e. there has been roundrobin competition) then the natural way to rank objects would be according to their scores si = P n=1aij. If, on the other hand, the outcome matrix lacks this symmetry, it is reasonable to suspect that ranking by score is not necessarily the best possible choice. (Contrary to common mathematical usage, we will often use the word tournament when referring to this asymmetric case; when we wish to emphasize the possible lack of symmetry, we may use the phrase generalized tournament.) A wide variety of methods have been proposed for ranking generalized tournaments. (See, for example, [9, 10].) In 1929, Zermelo [33] derived the functional

Slowly-migrating transition layers for the discrete Allen-Cahn and Cahn-Hilliard equations

It has recently been proposed that spatially discretized versions of the Alien-Cahn and Cahn-Hilliard equations for modelling phase transitions have certain theoretical and phenomenological advantages over their continuous counterparts. This paper deals with one-dimensional discretizations and examines the extent to which dynamical metastability, which manifests itself in the original partial differential equations in the form of solutions with slowly-moving transition layers, is also present for the discrete equations. It is shown that, in fact, there are transition-layer solutions that evolve at a speed bounded by C1 epsilon (1+C2/(n epsilon ))-C3n+C4 for all n>or=n0 and epsilon <or= epsilon 0, where 1/n is the spatial mesh size, epsilon is the interaction length, and n0 and epsilon 0 are constants.

Cell speed is independent of force in a mathematical model of amoeboidal cell motion with random switching terms.

In this paper the motion of a single cell is modeled as a nucleus and multiple integrin based adhesion sites. Numerical simulations and analysis of the model indicate that when the stochastic nature of the adhesion sites is a memoryless and force independent random process, the cell speed is independent of the force these adhesion sites exert on the cell. Furthermore, understanding the dynamics of the attachment and detachment of the adhesion sites is key to predicting cell speed. We introduce a differential equation describing the cell motion and then introduce a conjecture about the expected drift of the cell, the expected average velocity relation conjecture. Using Markov chain theory, we analyze our conjecture in the context of a related (but simpler) model of cell motion, and then numerically compare the results for the simpler model and the full differential equation model. We also heuristically describe the relationship between the simplified and full models as well as provide a discussion of the biological significance of these results.

Neighborhood monotonicity, the extended Zermelo model, and symmetric knockout tournaments

In this paper, neighborhood monotonicity is presented as a natural property for methods of ranking generalized tournaments (directed graphs with weighted edges). An extension of Zermelos classical method of ranking tournaments is shown to have this property. An estimate is made of the proportion of ordered pairs that all neighborhood-monotonic rankings of symmetric knockout tournaments have in common. Finally, numerical evidence for the asymptotic behavior of the extended Zermelo ranking of symmetric knockout tournaments is presented.

A continuous-time model of centrally coordinated motion with random switching

This paper considers differential problems with random switching, with specific applications to the motion of cells and centrally coordinated motion. Starting with a differential-equation model of cell motion that was proposed previously, we set the relaxation time to zero and consider the simpler model that results. We prove that this model is well-posed, in the sense that it corresponds to a pure jump-type continuous-time Markov process (without explosion). We then describe the model’s long-time behavior, first by specifying an attracting steady-state distribution for a projection of the model, then by examining the expected location of the cell center when the initial data is compatible with that steady-state. Under such conditions, we present a formula for the expected velocity and give a rigorous proof of that formula’s validity. We conclude the paper with a comparison between these theoretical results and the results of numerical simulations.

Blowup in a mass-conserving convection-diffusion equation with superquadratic nonlinearity

A nonlinear convection-diusion equation with boundary condi- tions that conserve the spatial integral of the solution is considered. Previous results on nite-time blowup of solutions and on decay of solutions to the corresponding Cauchy problem were based on the assumption that the nonlin- earity obeyed a power law. In this paper, it is shown that assumptions on the growth rate of the nonlinearity, which take the form of weak superquadraticity and strong superlinearity criteria, are sucient to imply that a large class of nonnegative solutions blow up in nite time.