Applied Mathematics Colloquium
Please note Applied Mathematics Colloquium and the Lefschetz Seminar have traded days, rooms and times for this week only ! |
Undergraduate Student Seminar
The Fluids, Thermal and Chemical Processes Group
of
The Division of Engineering
and
The Center for Fluid Mechanics
Abstract: Miniaturization of bioanalytical devices is of high interest for counteracting bioterrorism, enhancing drug discovery, and speeding biochemical research. Microchip-based analytical techniques have allowed increases in the complexity of microanalytical systems, primarily through the development of more sophisticated geometries.
Nanostructured and nanoporous materials are beginning to find application for miniaturized analytical systems and will doubtlessly find increased use in the future. Here, applications will be presented in which nanoporous polymer materials in different configurations create new capabilities for manipulation of biological samples in microanalytical systems.
We will first present novel techniques for high-pressure (10-300 bar) microfluidic control. In-situ laser-induced polymerization is used to fabricate fluorinated nanoporous polymers inside silica microchannels, whose actuation (actively voltage-addressed or passive) can be used to rout high pressure flows in a wide variety of solvents for chemical synthesis or chromatography. While any number of valving approaches have been proposed and implemented in microanalytical systems, we have developed the unique capability to valve high pressure flows with both aqueous and organic analytes. Application of these flow control elements is demonstrated in an integrated microchip HPLC system that can perform on-chip injections of 200 pl samples and has performed simple HPLC protein separations in 40 s, using standard water-ACN mixtures and ion-pairing reagents.
We will also present laser-fabrication techniques for photopatterning nanoporous dialysis membranes inside microchannels. Zwitterionic, nanoporous polymers with 5-50 \mu m thicknesses and 5-150 kDa molecular weight cutoff are fabricated inside microchannels using shaped laser light and used for protein concentration or counterflow dialysis. These membranes allow buffer ion flow and, in the presence of electric fields, can be used to trap proteins at the interfacewith standard pinched electrokinetic injection schemes, proteins can be concentrated by 2-4 orders of magnitude. In the absence of electric fields, these membranes can be used for desalting or other dialysis applications in counterflow configuration. The ability to pattern multiple membranes with arbitrary pore size opens up exciting possibilities for rapid dialysis of nascent analytes on chip.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: Initially we study some weak Riemannian metrics on spaces of regular smooth curves in the plane, viewed as orbit spaces under the group of diffeomorphisms of the circle acting as reparameterizations. In particular we investigate the metric for a constant $A> 0$: $$ G^A_c(h,k) := \int_{S^1} (1+A\kappa_c(\th)^2)\langle h(\theta),k(\theta) \rangle |c'(\theta)|\,d\theta $$ where $\kappa_c$ is the curvature of the curve $c$.
Big surprise: For $A=0$ this induces 0 as geodesic distance function on the orbit space. But for $A>0$ it induces a point separating metric. We give some estimates for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions.
The big surprise for $A=0$ remains true on the space $Imm(M,N)/Diff(M)$ of all submanifolds of type (compact) $M$ in a Riemannian manifold $(N,g)$, for the analog of the metric above. The second part of the talk will concentrate on the higher dimensional situation. This is joint work with David Mumford.
Special Lefschetz Center for Dynamical Systems Seminar
Please note Applied Mathematics Colloquium and the Lefschetz Seminar have traded days, rooms and times for this week only! |
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: In this talk I will describe a top-down approach for image segmentation how it is learned automatically from training images, and how it can be effectively combined with bottom-up segmentation methods. The approach uses sto red class representations, learned automatically from unsegmen ted training images, to detect and recognize object parts. These are then used to guide the segmentation process. This top-down approach contrasts with the common bottom-up approach to recognition and segmentation, which assumes that image segmentation is performed first, followed by object recognition. The method successfully addresses the main difficulty of top-down segmentation: high variability in the shape and appearance of objects within a specific class. Computational experiments with several object classes show that this method leads to markedly improved results, and can deal with significant variation in shape and background. I will conclude by demonstrating how the top-down approach can be effectively combined with bottom-up segmentation in a way that integrates the relative advantages of both approaches. This combination can be applied to a range of existing bottom-up and top-down methods. If time permits, I will also discuss biological data relevant to the top-down model, as well as the potential use of this scheme to enhance recognition and classification.
Joint work with Shimon Ullman
Scientific Computing Seminar
Abstract: Aerodynamic simulations for complex applications are largely based upon (at best) second-order accurate methods, however, the potential exists for higher- order discretizations to greatly improve the efficiency of CFD. To achieve a reliable higher-order method for industrial-strength problems, a variety of challenges must be addressed, for example: the efficient solution of the discrete equations; a robust limiting strategy for shocks; an $hp$-adaptive algorithm to automatically and robustly resolve the solution. In this talk, we consider the use of discontinuous Galerkin (DG) approximations for the development of a higher-order solver for the compressible Reynolds-Averaged Navier-Stokes equations. We present a full-approximation, p-multigrid algorithm that uses a hierarchical polynomial basis to efficiently construct lower-order discretizations. On a given p-level, a line block-Jacobi smoother is used in which a unique set of lines are formed by grouping tightly- coupled elements with the coupling determined by the flux through an element face. Fourier analysis of the smoother shows that it is stable for all orders of approximation without the need for multi-staging. We demonstrate that asymptotic residual convergence rates for the algorithm are insensitive to the polynomial order though are still dependent on the mesh spacing. However, for smooth flows, solution errors achieve a priori estimates within a few multigrid cycles though a finite residual remains. We also construct an adjoint-solution technique which is guaranteed to have the same asymptotic convergence rate as the flow solver, and demonstrate the potential computational advantages in simultaneously solving the primal and dual problems.
<--- 2004 Index