Lefschetz Center for Dynamical Systems Seminar
Abstract: In the focusing problem, we solve the initial value problem for the porous medium equation with an initial distribution whose support lies outside a compact set $ K $. At a finite positive time $ T $ the exterior of the support of the solution shrinks to a point, and we are interested in the asymptotic of the process in the neighborhood of the focusing point for times near $ T $. If $ K $ is radially symmetric the asymptotic behavior is described by a one parameter family of self-similar solutions. These radial solutions are unstable to non-radial perturbations and there is a sequence of symmetry breaking bifurcations leading to new families of non-radial self-similar solution. We discuss this bifurcation structure as well as a class of non-radial non-self-similar patterns.
Brown University Center for Statistical Sciences Seminar
Public Health Sciences, Wake Forest University School of Medicine | |
*Reception following seminar at 167 Angell St., 2nd floor conference room. Sponsored by the Mollie B. Mandeville Lectureship and The C. V. Star Foundation Lectureships Fund |
Abstract: The elderly population in the U.S. is expected to double in size over the next 25 years making longitudinal health studies of this population of increasing importance. The degree of loss to follow-up in studies of the elderly, which is often due to elderly persons being incapable of remaining in the study, entering a nursing home or dying, make longitudinal studies of this population problematic. We propose a latent class model for analyzing multiple longitudinal binary health outcomes with multiple cause non-response when the data are missing at random (MAR) and a non-likelihood based analysis is performed. We extend the estimating equations approach of Robins et al. (1995) to latent class models by re-weighting the multiple binary longitudinal outcomes by the inverse probability of being observed. This results in consistent parameter estimates when the probability of non-response depends on observed outcomes and covariates (MAR) assuming the model for non-response is correctly specified. We extend the non-response model so that institutionalization, death, and missing due to failure to locate, refusal or incomplete data each have their own set of non-response probabilities. Robust variance estimates are derived which account for the use of a possibly misspecified covariance matrix, estimation of missing data weights and estimation of latent class measurement parameters. This approach is then applied to a study of lower body function among a subsample of elderly participating in the six year Longitudinal Study of Aging (LSOA).
Center for Fluid Mechanics Seminar
Abstract: Peskin has introduced the virtual boundary method for numerical simulation of flow patterns around mitral heart valve. Peskin's method represents a surface within a flow field via a forcing term added to governing Navier-Stokes equations. The method imposed the forcing term only at the surface points. Investigations of the effect of solid bodies immersed within a flow field suggest that in implementing the virtual boundary technique the forcing term should be applied at the boundary and interior points of the body. As a result, an unphysical flow is developed inside the "body". The "flow field" developing inside the cube is of unphysical nature, and its assignment is effectively to enforce impermeability (zero velocity) conditions on the "body".
Following the spirit of the virtual boundary technique recently introduced by Goldstein, Handler and Sirovich, we developed a numerical scheme for simulating the turbulent flow around a cube placed on a channel wall. The scheme could be implemented for direct numerical simulations (DNS), as well for Reynolds-averaged Navier-Stokes (RANS) simulations. The flow around a cube placed in a channel has been thoroughly studied experimentally has been used as the benchmark for validation CFD codes and assessment of turbulence models. This case is a typical example of flows with a strong intrinsic three-dimensionality, which involves complex interaction between major separation vortices and a horseshoe, which is developed behind the cube. Computer animation based on numerical results clearly shows the complicated features related to the formation of horseshoe structures.
Stochastic Systems Seminar
Abstract: Real-time systems are computer and communications systems in which the tasks using the system have explicit timing requirements ( due dates, or deadlines). Important application areas include multimedia communications (video conferencing, speech and image recognition), parallel computing (message passing) and embedded real-time control systems.
Traditional deterministic analysis handles almost exclusively the case of periodic arrivals and assumes worst-case execution times. In fact, real-time systems can exhibit substantial variability in arrival of tasks and their work requirement. To avoid these shortcomings, a stochastic approach, Real-Time Queueing Theory, has been proposed and developed. The key feature of this approach is that it keeps track of the dynamically and stochasticallly evolving lead-time profiles of customers. This leads to models with infinite dimensional state spaces, making their analysis extremely difficult. A major simplification can be achieved by considering the heavy-traffic limit of the processes involved.
In my talk I would like to focus on two methods for obtaining heavy traffic limits, namely the semigroup approach, and an approach using a Functional Central Limit Theorem (FCLT).
For the case of EDF and FIFO service disciplines, the FCLT approach seems to be more natural and to give stronger results. The semigroup approach involves lengthy calculations and its results are limited to markov queues.
Yet, in my opinion, the semigroup approach holds some significant promises and advantages. Firstly, some service disciplines like PS (processor sharing) seem to be unyielding to the FCLT approach, whereas the semigroup approach seems to be straightforward. Second, an important question in these investigations is not only to identify the heavy-traffic limit, but also to give an idea of the convergence rate; again, the semigroup approach seems to be better suited for this task.
Brown Applied Mathematics Pattern Theory and Vision Seminar
**Please Note This Seminar Will Be Held In The MacMillan Building
The Brain Science Program
and
The Division of Applied Mathematics
Present
**Please Note Change of Location for This Week Only |
Abstract: A major unsolved problem regarding the organization of neocortex concerns how bottom-up, top-down, and horizontal interactions are organized within layers to generate adaptive behaviors. This talk describes a neural model of how these interactions help visual cortex to realize: the binding process whereby cortex groups distributed data into coherent object representations; the attentional process whereby cortex selectively processes important events; and the developmental and learning processes whereby cortex shapes its circuits to match environmental constraints. The model is used to simulate neurophysiological, anatomical, and psychophysical data about how the LGN and areas V1 and V2 are organized and interact.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: Protein channels conduct ions $\(Na^+, K^+, Ca^{++},} {\rm and } Cl^-)$ through a narrow tunnel of fixed charge (`doping') thereby acting as gatekeepers for cells and cell compartments. Hundreds of types of channels are studied every day in thousands of laboratories with the powerful techniques of molecular biology because of their biological and medical importance: a substantial fraction of all drugs used by physicians act directly or indirectly on channels. The atoms of channels can be manipulated one at a time and the location of every atom can be determined within 0.3 \AA in a few cases; more to come. They are ideal objects for mathematical and computational investigations.
Ionic channels are `holes in the wall' that use the simple physics of electrodiffusion to perform these important tasks. Computing the movement of spheres through a `hole in the wall' should be easier than computing most other biological functions, yet it is nearly as important as any, from a medical and technological point of view.
The function of open channels can be described if the electric field and current flow are computed by the Poisson-Drift-Diffusion equations (called {\it PNP}, for Poisson Nernst Planck, in biology) and the channel protein is described as an invariant arrangement of fixed charges --not as an invariant potential of mean force or set of rate constants, as is done in the chemical and biological tradition. The Poisson-Drift-Diffusion equations describe the flux of individual ions (each moving randomly in the Langevin trajectories of Brownian motion) in the mean electric field. They are nearly identical to the drift diffusion equations of semiconductor physics used there to describe the diffusion and migration of quasi-particles, holes and electrons. They are closely related to the Vlasov equations of plasma physics.
\it {PNP} fits a wide range of current voltage (\it {I-V}) relations--whether sublinear, linear or superlinear-- from 7 types of channels, over +/-180 mV of membrane potential, in symmetrical and asymmetrical solutions of 20 mM to 2 M salt. The \it {I-V} relations of the gramicidin channel can be predicted directly from its structure, known as \it{NMR}, using an independently measured diffusion coefficient and no adjustable parameters. Porin channels with known structure have been studied, and parameter estimates (in mutations also of known structure) are surprisingly close to those predicted (i.e., within 7%). Selectivity has been studied extensively in the calcium release channel of cardiac muscle: \it{I-V} relations in Li^+, K^+, Na^+, Rb^+, and Cs^+ and their mixtures can be explained with a few invariant parameters (of reasonable value) over the full range of concentrations and potentials. Complex selectivity properties of channels are easily explained: the anamalous mole fraction effect in K^+ and {\it L}-type Ca channels arise naturally as a consequence of binding. Indeed, the selectivity of the {\it L}-type calcium channel can be predicted quantitatively if permeating ions are treated as finite objects with the entropy and electrostatic energy of spheres. {\it L}-type Ca channels are of particular clinical importance because they control the heart beat and is the target of calcium channel blockers, drugs taken by a substantial fraction of the population.
Taken together, these results suggest that open ionic channels are natural nanotubes, dominated by the enormous fixed charge lining their walls (~5 M, arising from 1 charge in 7x10\AA). Physical chemists (e.g., Henderson, Blum, and Lebowitz, 1979, J. Electronal. Chem. 102: 315) have shown that highly charged systems are dominated by their mean electric field and the changes in the shape of the electric field. Atomic detail is unexpectedly unimportant because correlation effects are small. Biologists and biochemists have traditionally focused on correlation effects and more or less ignored the electric field. Thus, the success of the Poisson-Drift-Diffusion equations and the predominant role of the electric field has been a surprise to biologists, although the role of the electric field is hardly a surprise to physicists who have long understood the importance of the electric field in semiconductors and ionized gases, i.e., plasmas.
The role of the electric field is not so clear cut in biology. Protein structure, protein folding, nucleic acids (i.e., DNA), and drug binding are systems of the greatest importance to biological research and computational biology, and are the recipients of considerable funding, yet traditional analysis more or less ignores the electric field. Traditional analysis never treats the electric field self consistently, as a result of charge, in the presence of flux.
An opportunity exists to apply the well established methods of computational physics to the central problems of computational biology. Many of those methods have not yet been used to analyze other biological systems. Perhaps they should be: the application of even the lowest resolution techniques involving the Poisson-Drift-Diffusion equation has transformed the study of channels. It is likely that application of higher resolution methods, like self-consistent Monte Carlo/Molecular dynamics, would have a comparable effect on other areas of computational biology.
In my opinion, the plasmas of biology need to be analyzed like the plasmas of physics. The mathematics of semiconductors and ionized gases should be the starting point for the mathematics of ions and proteins, for the analysis of protein structure, protein folding, nucleic acids (i.e., DNA), and the binding of drugs to proteins and nucleic acids.
PDE Seminar
Abstract: Among all sets of a given volume, the ball has the smallest possible surface area. Balls have many other extremal properties; for instance, a particle performing Brownian motion will "on average" take more time to leave a ball than it would take to leave any other set of the same volume.
In this talk, I will discuss a useful tool for proving such statments: a comparison principle for multiple integrals that was established by Brascamp, Lieb, and Luttinger (BLL) in 1977. I would like to explain why it is true, and how it relates to isopermetric inequalities. Certain aspects of the BLL inequality are not well understood, including its cases of equality and possible extensions to curved spaces. I will conclude with some recent results in that direction.
Department of Mathematics Colloquium
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