Lefschetz Center for Dynamical Systems Seminar
Abstract: Gap solitons are a species of travelling wave that propagate through specialized Bragg Grating optical fibers. They have the enticing property of propagating at any speed between zero and the speed of light. Technical difficulties have prevented experimentalists from actually creating light pulses with zero velocity, which could someday be useful in optical communications devices.
I have been investigating an alternative approach, in which specialized defects are designed to "trap" the light at a designated location. I study this from a variety of angles. First, through extensive simulations of the interaction of the solitons and defects, I develop a theory that predicts under what circumstances light energy is trapped by the defect. Second, I derive finite dimensional (ODE) models for related systems which we can analyze more completely using techniques from dynamical systems theory.
Joint work with Michael Weinstein, Phil Holmes, Dick Slusher.
Brown University Center for Statistical Sciences Seminar
Department of Mathematics and Statistics, University of Massachusetts
*Reception following Seminar - 167 Angell Street, 2nd floor Conference Room.
For more information, please contact Jason Roy at 863-9559, e-mail email@example.com or Ilana Gareen at 863-1758, e-mail firstname.lastname@example.org
Abstract: We apply the profile likelihood method based on the multinomial distribution to inference on the accuracy of a diagnostic test. The methods are applicable to ordinal rating data and accuracy is assessed using the Area Under the Receiver Operating Characteristic (ROC) curve. Simulation results suggest that the derived confidence intervals have acceptable coverage probabilities, even when sample sizes are small and the diagnostic tests have high accuracies. The methods extend to stratified settings and situations where the ratings are correlated. We illustrate the methods using data from a clinical trial on the detection of ovarian cancer.
Center for Fluid Mechanics and The Division of Engineering
Stanford University, Stanford, CA
Abstract: Within the last five years, the use of strong video microscopy to visualize DNA chains in flow as pioneered by Chu and co-workers is really revolutionizing the field of dilute solution rheology. Rather than simply postulating coarse-grained models for a chain in various "strong" or"fast" flow fields (and then examine the resulting stresses in comparison to measured values), researchers can now directly examine the configurational statistics of a molecule in flow far from equilibrium. Combining these experimental studies with Brownian dynamics where an assortment of molecular models can be examined, provides and extremely powerful tool to probe the physics which must be included in a model to have it faithfully capture the molecular dynamics. Note in this context, that Brownian dynamics simulations are playing an increasing role in elucidating the physics necessary to include in the microscopic models that are critical to the development of any numerical scheme for calculating viscoelastic flows. Thus, micro-macro slow solvers (where a microscopic model for the polymer dynamics is directly coupled to a macroscopic numerical scheme) are increasingly the methods of choice for calculating the flow of polymeric liquids. In previous work, Brownian dynamics simulations have been extremely useful in our understanding of polymer stretch and recoil in extensional flows, with in extensional flows, with now a growing body of literature on the subject. There is less available for understanding how shearing flows effect single chain dynamics. Simple shearing flow is almost a universal local flow condition for polymer chains in flows which are parallel or nearly parallel on the scale of the polymer molecule, e.g. all viscometric flows.
In this presentation, I will first present the results of large-scale Brownian dynamics simulations which have been used to examine the recent experimental findings of Smith et al. (Science 283 (99)) who studied a single DNA molecule in steady shear flow. We have used a series of different models - the Kramer's freely jointed chain, the Wormlike spring chain, the Inverse Langevin chain, the Hookean dumbbell and the Rouse chain - to understand the experimental findings. Moreover, I will demonstrate that the dynamics of these single chains is quite remarkable particularly at large flow strengths. I will then present joint work with the Chu group which includes a combined experimental/Brownian dynamics study of the start-up of steady shear flow. I will demonstrate how the salient features of polymer dynamics in a transient flwo can be captured by the aforementioned chain models, but only if sufficient internal modes and/or flexibility are included in the description. Moreover, I will present experimental work where we demonstrate how increasing the concentration of surrounding molecules affects the dynamics of a single chain and use this to motivate a discussion of how the flow dynamics of single molecules bears directly on the material properties of non-dilute polymer solutions. We include mechanical measurements of the rheology of DNA solutions which we can predict from the dynamic simulations. Finally, I will present our latest work in the area which includes examining DNA chains in mixed flows (i.e. planar flows with varying ratios of extension and vorticity) as well as shear flows near and between rigid impermeable boundaries. The latter work includes studies of DNA chain scission in contraction flows as a preparatory step for sequencing.
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: Recently an interesting modification of the Navier-Stokes equations was introduced by Holm, Marsden, and Ratiu (1998). These modifications filter the fluid motion equations that occurs below some length scale, alpha, which is a parameter in the model. Formally, alpha is a tensor that represents the Eulerian mean displacement fluctuation covariance. Practically it represents the smallest scale in the solution, beyond which the dynamics may not matter.
These new sets of equations are more computable than the Navier-Stokes equations. For example, Foias, Holm, and Titi show that these equations have a finite dimensional attractor and have proved existence and uniqueness in the strong form for periodic domains, something which has not been proved for the Navier-Stokes equations.
Unlike many other averaged versions of the Navier-Stokes equations, the Navier-Stokes-alpha equations preserve the basic transport structure of the Navier-Stokes equations. For example, they have Kelvin circulation theorems and conserve energy in an $H^1$ norm. The Navier-Stokes-alpha equations regularize the Navier-Stokes equations through nonlinear dynamics, not through added dissipation.
In this talk I discuss the related equations, the shallow-water-alpha and the primitive-alpha equations that are used in Geophysical Fluid dynamics, including some numerical properties.
Special Stochastic Systems Seminar
Abstract: This talk concerns policy synthesis in large queuing networks. The results provide answers to the following questions:
(i) It is well-known that an understanding of variability is important in the determination of safety stocks to prevent unwanted idleness. Is this the only use of high-order statistical information in policy synthesis?
(ii) Will a translation of an optimal policy for the deterministic fluid model (in which there is no variability) lead to an allocation which is approximately optimal for the stochastic network? If so, what is the `regret'?
(iii) Where are the highest sources of sensitivity in network control?
A sensitivity analysis of an associated fluid-model optimal control problem' provides an exact dichotomy in (ii). If an optimal policy for the fluid model is `maximally non-idling', then variability plays a small role in control design.
If this condition does not hold, then the `gap' between the fluid and stochastic optimal policies is exactly proportional to system variability. However, sensitivity of steady state performance with respect to perturbations in the policy vanishes with increasing variability.
Department of Mathematics Colloquium
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