Seminar on Nonlinear Waves
Brown University Center for Statistical Sciences Seminar
Abstract: A key determinant of whether a test for a randomized treatment effect derived from a mis-specified proportional hazards regression model is valid is whether the distribution of censoring depends on both treatment group and covariates. If censoring is independent of either treatment group or the covariates, then such tests are asymptotically valid whether or not the model is mis-specified (Kong & Slud, 1997). However, if censoring depends on both treatment group and covariates, the size of logrank and other tests arising from Cox's (1972) partial likelihood function can be seriously distorted (DiRienzo/&Lagakos, 1998). We propose a corrected test for these settings which replaces the at-risk indicator of risk-set membership by a continuous function taking values in the unit interval that is independent of treatment group, thus leading to an asymptotically valid test. Simulations indicate that the bias-corrected test can work well when the dependence of censoring on treatment group and covariates is not modeled exactly. An example from an AIDS clinical trial illustrates the technique.
Lefschetz Center for Dynamical Systems Seminar
Abstract: In this talk I will overview the theory of translational and oscillatory instabilities of solitons in models of nonlinear optics. In particular, I discuss oscillatory instabilities of solitons in the parametrically driven, damped nonlinear Schr\"{o}dinger equation. Using the methods of integral transforms, eigenfunction expansions and asymptotic approximations, I derive reduced amplitude equations (normal forms) for Hopf and Hamiltonian Hopf bifurcations of soliton instabilities. In the undamped case, the instability arises due to resonance of two internal oscillation modes and results in evolution of unstable solitons either into decaying breathers or into radiating solitons of growing amplitudes. Formation of stably oscillating solitons is supported by sufficient damping in the parametrically driven system.
Special Applied Mathematics Colloquium
Abstract: I will discuss the dynamics of multiple phenotypes of a species in a continuous but heterogeneous environment where the phenotypes are characterized by their diffusion rates. I will discuss conditions under which spatial or temporal heterogeneity leads to the selection of slow diffusion.
Joint Seminar, Center for Fluid Mechanics and Division of Engineering
Abstract: Boundary-layer stability experiments are conducted in the ASU Unsteady Wind Tunnel on a 45-degree swept airfoil. The pressure distribution is designed to make the boundary layer subcritical to T-S waves (streamwise instabilities) but very unstable to crossflow waves. The crossflow waves can be both stationary and traveling. The surface is polished to 0.2 micron rms in order to minimize background induced disturbances. Uniform stationary crossflow vortices are generated by placing specific arrays of micron-sized roughness elements near the attachment line in order to develop single- or multiple-mode disturbances. Roughness spacing and Reynolds number are varied to examine the behavior of these modes and their harmonics. Widely spaced roughness (at three times the most unstable wavelength) show growth of the first nine harmonics. The measurements show a clear nonlinear distortion of the mean flow and saturation of the stationary structure at considerably lower amplitude than predicted by linear theory. Comparisons with nonlinear PSE calculations show outstanding agreement in the growth rate and amplitude saturation when curvature is taken into account. It is also shown that certain roughness spacings inhibit the growth of the most unstable modes and transition can be moved beyond the smooth case. These elements are typically 6–18 $\mu$m high and 2–4 mm in diameter and are placed near the attachment line of a swept airfoil where the displacement thickness is of the order of 250 $\mu$m.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: A detection algorithm which employs a version of the Hough transform based on trained local features will be described. The features are defined in terms of flexible local conjunctions of edges. At reference scale stable local features are identified for each object class at specific locations.
A parallel implementation of this algorithm through a sequence of retinotopic feature detector layers will be described. The main point will be to use the same architecture to detect any object whose representation in terms of features/locations is evoked in a main memory module.
I will present analogies between this architecture and known functions of the visual system, in particular some experiments on visual selection.
Brown Analysis Seminar
Applied Mathematics Colloquium
Scientific Computing Seminar
Abstract: Spectral methods are popular because of their exponential convergence properties for continuous smooth functions. However, spectral methods cannot accurately recover piecewise smooth functions due to the spurious oscillations that develop near the jump discontinuities and the overall convergence rate reduction to $O(\frac{1}{N})$.
Much work has been devoted to this problem, resulting in reconstruction methods that successfully eliminate the Gibbs phenomenon and recover piecewise smooth functions with algebraic and even spectral accuracy. These reconstruction methods require apriori knowledge of the locations of the jump discontinuities, and hence the detection of these edges is essential in obtaining an overall spectrally accurate solution.
This talk focuses on recent developments in the edge detection method developed by Gelb and Tadmor (1998). The edge detection method uses smooth {\em concentration kernels}, $K(t) = K_\epsilon(t)$, depending on a small parameter $\epsilon$ and characterized by \[K_\epsilon\*f(x) \longrightarrow [f](x), \hspace{.2in} \epsilon \rightarrow 0,\] where $[f](x)$ denotes the local jump function. A nonlinear enhancement (Gelb/Tadmor 1999) further `pinpoints' the edges by enhancing the separation in scales that the concentration kernel yields. Once the edges of these discontinuities are identified, we can utilize a reconstruction method to accurately recover the piecewise smooth function $f(x)$.
Most recently a two-dimensional enhanced edge detection method has been developed, as well as a two-dimensional recovery method for piecewise smooth functions, based on the one dimensional results of Gottlieb and Tadmor (1984). The combination of these two methods yields a fully automated and robust two-dimensional reconstruction scheme. This new edge detection and recovery scheme has many applications, and here we provide a demonstration in solving hyperbolic conservation laws.
PDE Seminar
Department of Mathematics Colloquium
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