Jan. 10

The long time numerical approximation of the parabolic $p-$Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are {\em uniform} for $t\in [0,\infty)$ are established in the $L^2$, $W^{1,p}$ norms for the backward Euler and the Crank-Nicolson schemes with the Finite Element Method. This result extends the existing uniform convergence results for {\em exponentially} contractive semigroups generated by some {\em semi-linear} systems to {\em non-exponentially} contractive semgroups generated by some {\em quasi-linear} systems.

Jan. 31

In the move towards mass customization, many companies, especially in the telecommunications and electronics industries, have adopted assemble-to-order (ATO) practices. This means keeping stocks of key components and submodules but assembling final products only in response to customer orders. This allows the companies to compete in fast-changing markets, while avoiding costly piles of unsold products. (The automobile industry is now catching up in this direction.) In such systems, as in any multi-item inventory system, a customer order typically consists of several different items in different amounts. The order-based performance measures, such as the average number of customer requests backlogged and the probability of satisfying an arbitrary customer order within a prespecified time window (order fill rate), are important measures of customer satisfaction. However, due to the demand correlation across items, it is difficult to predict the performance as a function of stock levels. In this talk, we will present some recent development in modeling the ATO systems and the corresponding solution techniques for performance evaluation and optimization. We will also discuss the insights gained from the models. Along the presentation, we will point out directions for future research.

Feb.18

We review some aspects of viscous fingering in a Hele-Shaw cell that at first sight appear to defy intuition. These include singular effects of surface tension relative to the corresponding zero-surface-tension problem both for the steady and unsteady problem. This also includes disproportionately large influence of small effects like local inhomogeneity of the flow-field near the finger tip, or of the leakage term in boundary conditions that incorporate realistic thin-film effects. Through simple explicit model problems, we demonstrate how such properties are not unexpected for a system approaching structural instability or ill-posedness.

Feb.28

We derive an exact formula for solutions to the Stokes equations in the half space with an external force. This formula is used to establish local and global existence and uniqueness for large and small data respectively in a suitable Besov space for solutions to The Navier-Stokes equations.

Feb.4

We consider an initial-boundary value problem for a wave equation in high dimensions with a nonlinear damping term. We establish the existence and uniqueness of a global solution by using acompactness method and by exploiting the monotonicity property of the nonlinearity.

Feb.8

The enhancement effect of wind or fluid motion on the rate of chemical reactions, and front propagation in general, is a well known phenomenon, that occurs in many applications ranging from combustion to biology. We will review some previous mathematical results regarding front propgation in the presence of advection based on homogenization theory and existence of travelling waves for some classes of flows. However, these methods do not provide explicit bounds on the propagation speed in terms of the advecting flow, and are applicable for a limited range of flows. We will introduce an unambiguous way to measure the reaction rate in the situations when these approaches may not work. We will describe bounds on the reaction rate in terms of the magnitude, geometry and scale of oscillations of the advecting flow. In particular this will allow us to estimate the speed of the travelling fronts in shear and vortical flows.

Feb.11

We consider wave propagation in multiscale random media. Our objective is to characterize how a pulse propagating through the random medium is affected by the medium fluctuations. Multiscale or fractal random media are used to model for instance the heterogeneous earth and the turbulent atmosphere. For a class of fractal random media defined in terms of fractional Brownian motion we show how an acoustic wave pulse interacts with the medium fluctuations. The modification in the pulse shape depends on the roughness of the medium and can be described in a **deterministic** way when the pulse is observed at its **random** arrival time.

Mar. 10

Seeing is not as simple as it looks. Each glance engages billions of neurons, trillions of synapses, and roughly half of the brain's cortex. Why? Because vision does not simply copy the visual world, but instead constructs the visual world with all its depth, colors, motions, and objects. Sophisticated rules govern the constructive processes of vision and our other senses, rules now being studied in the cognitive and neural sciences. In this talk I show many visual effects which illustrate the constructive nature of vision. I then discuss current formal models of visual processing, and highlight the important role and opportunities for mathematics in modern visual science.

Mar. 15

The hydrostatic shape, transient deformation, and asymptotic shape of a small liquid drop with uniform surface tension adhering to a planar wall subject to an overpassing simple shear flow are studied under conditions of Stokes flow. Families of hydrostatic shapes are computed using an iterative finite-difference method. The transient deformation of a drop subject to a suddenly applied simple shear flow is computed for a range of capillary numbers using a boundary integral method.

Mar. 20

The mechanisms involved in the complex biological processes that control wound healing are still poorly understood. A new approach which develops plausible models, based on known biology, can suggest possible mechanisms and pose relevant biological questions. I shall describe how such models are derived from known biology, present two specific models for full depth dermal wounds and for epithelial wounds and present some of the clinical implications. (i) In the case of dermal wounds, fibroblast traction forces play a crucial role. They are key players in distorting the various substrata and in reorganising the extracellular medium in the early stages of the wound healing. Our model specifically addresses the wound contraction dynamics and shows how fibroblast traction forces and the elastic and plastic response of the extracellular matrix can account for the different experimental phases of the wound closure as a function of time. Further it provides a quantification of the residual strain and stress from the plastic response and remodelling of the extracellular matrix which could help to characterise the scar tissue formation in dermal wound healing. (ii) The study of epidermal wounds lets us consider wound healing independent of wound contraction and other complexities associated with full depth wounds. In epidermal wound healing in mammals cells spread across the surface of the wound. The stimulus for increased epidermal cell division is uncertain. Our model suggests that a single chemical with a simple regulatory effect can account for the healing of epidermal wounds. The results compare well with experimental data. I shall then make certain predictions as to healing time as a function of wound geometry. Epidermal wound healing in embryos is startlingly different to that in adults. I shall describe some recent remarkable experiments which highlight the differences and touch on a model, based on the experiments, to show how it can help explain the observed results and resolve a biological paradox.

Mar. 13

This talk will address the interactions between flow and microstructure of complex fluids, such as liquid crystalline polymers (lcps). These materials have molecular anisotropy, which is responsible for pure material phenomena such as the isotropic-to-nematic phase transition. When exposed to simple flows such as shear and elongation, many other remarkable structures and behavior are observed. We will discuss the present mesoscale models for these materials, and then elicit some information from them, even perhaps of relevance to observed patterns and transition phenomena. The talk is based on joint work with Qi Wang, IUPUI and Hong Zhou, UC-Santa Cruz.

Mar. 3

The Camassa-Holm equations may be viewed as an averaged version of the Navier-Stokes equations. This averaging decomposes material trajectories into a mean path plus a rapidly varying deviation and then makes an approximation to obtain an equation governing the mean path. Thus, the time dependent Camassa-Holm equations should reflect the dynamical properties of the original turbulent flow while at the same time suppressing irrelevant small scale motion. In particular, it is hoped that the degrees of freedom for the Camassa-Holm equations would be the same as for the Navier-Stokes equations. This talk will present some recent numerical computations of the energy spectrum and compare estimates for the dissipative cutoff of the energy cascade with the degrees of freedom for a turbulent flow as given by Kolmogorov.

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