Brown University Graduate School
Dissertation Defense Information Seminar
Brown Analysis Seminar
Abstract: We establish in a setting of harmonic analysis precise relationships between combinatorial measurements and Orlicz norms. These relationships further extend and sharpen prior results concerning extensions of the Littlewood 2n/(n+1)-inequalities, the n-dimensional Khintchin inequalities, and the Kahane-Khintchin inequality.
Center for Statistical Sciences Seminar
Abstract: Functional neuroimaging studies attempt to identify spatially localized brain regions that drive the execution of experimental tasks targeting, for example, behavior, cognition, or emotion. These studies may also provide neural representations of the pathophysiology associated with psychiatric, neurological, and addiction disorders. There are numerous challenges in analyzing functional neuroimaging data. Neuroimaging studies produce massive data sets comprised of serial scans on each subject, with each scan containing hundreds of thousands of spatially localized measurement sites (voxels). In this talk, we propose a Bayesian hierarchical model that accounts for spatial correlations between voxels, both within and between designated neuroanatomic structures, while also capturing temporal correlations between repeated scans for each subject. We apply this model to functional magnetic resonance imaging (fMRI) data from a study of inhibitory control among cocaine addicts. Our model provides a unified framework to detect localized (voxel-level) changes in inhibitory control related brain activity associated with cocaine addiction and its response to treatment, regional changes in brain activity, and measures of task-related functional connectivity.
Lefschetz Center for Dynamical Systems Seminar
Stochastic Systems Seminar
Abstract: Poincare Inequality on a compact Riemannian manifold is of the form: $$ \int f^2 dx < =C \int |df|^2 dx.$$ where the best constant is related to the bottom of the spectrum of the Laplace-Beltrami operator. Replace the Lebesque measure by a probability measure. The corresponding Poincare inequality reveals properties of the measure. Poincare inequalities hold for Gaussian measures, in which df needs to be interpreted as a gradient operator related to the Gaussian structure. The measure we are interested in is the Brownian bridge (BB) measure on the space of continuous curves over a Riemannian manifold. If the manifold is not simply connected the inequality is trivially false as first pointed out by L. Gross. A. Eberle showed that on a connected compact manifold, a Poincare inequality for BB measure fails for some Riemannian metric. We look into the case when the manifold is a hyperbolic space and give a positive result. This is joint work with X. Chen and B. Wu.
Center for Fluid Mechanics Seminar
Abstract:
The complex hydrodynamics of water entry are investigated experimentally for
low Froude numbers for spheres and ballistic projectiles. The cavity formed
by the projectiles can be readily altered as a function of projectile spin,
surface coating and density. For example, the splash and cavity dynamics of
the water entry process are highly altered for the spinning case compared to
impact of a sphere without spin. A transverse spinning motion induces a lift
force on the sphere and thus causes significant curvature in the trajectory of
the object along its descent, similar to a curve ball pitch in baseball. A basic
force model can be used to evaluate the lift and drag forces on the sphere
after impact; resulting forces follow similar trends to those found for
spinning spheres in oncoming flow, but are altered slightly as a result of
the subsurface air cavity.
Physical models are derived from the projectile behavior captured through
high-speed video image sequences. New phenomena have been witnessed via these
techniques including a wedge of fluid that crosses the cavity in the case of
transverse rotational velocity. The wedge formation is a result of the contact
line properties, surface roughness and spin rate. Non-rotating, spherical
projectiles impacting a water surface generate different cavity and splash
behaviors depending on the static wetting angle of the projectile surface, in
a range of low-moderate Froude numbers. The static contact angle made between
a liquid and a solid surface varies with surface coating and roughness, with
large angles for hydrophilic coatings and small angles for hydrophobic coatings.
For sufficiently hydrophilic surfaces it is possible to prevent cavity and
splash all together. The effect of surface coating on the impact of spinning
and non-spinning spheres will also be addressed.
Lefschetz Center Colloquium
Abstract: We consider one of the simplest possible models of non-equilibrium statistical mechanics: two coupled oscillators in contact with two Langevin heat baths. The twist is that one of the heat baths is at "infinite" temperature in the sense that no friction acts on the corresponding degree of freedom. We explore the question of the existence of a stationary state in this situation and its properties if it exists. In particular, we will see that the question "Is the corresponding degree of freedom at infinite temperature?" can have a surprising variety of answers.
Brown University Graduate School
Dissertation Defense Information Seminar
Center for Vision Research Seminar Series 2008-2009
Abstract: I will present evidence showing that individuals with large lesions of primary visual cortex can perform a broad range of skilled visually guided movements including scaling the hand in flight for the size of goal objects and avoiding obstacles in the workspace of the hand. In addition, I will present findings from functional magnetic resonance imaging (fMRI) showing that such patients continue to show task-related activation in dorsal-stream areas that have been implicated in the visual control of reaching and grasping. Taken together, these results suggest that extra-geniculostriate projections to the dorsal stream are capable of mediating the processing of object features such as size, shape, and orientation for the control of visually guided grasping.
Lefschetz Center for Dynamical Systems Seminar
Abstract: In order to describe slow modulations in time and space of spatially periodic solutions of pattern forming reaction-diffusion systems so-called phase diffusion equations and Cahn-Hilliard equations can be derived as formal approximation equations via multiple scaling analysis. If these phase diffusion equations are degenerate, there exist solutions showing a waiting time phenomenon. An example is the porous medium equation which can be derived as a degenerate phase diffusion equation for modulations of spatially periodic solutions of the real Ginzburg-Landau equation which have wave numbers close to the boundaries of the so-called Eckhaus-stable region. With the help of estimates between the formal approximations and the exact solutions of the original system we explain the extent to which these formal approximations are valid in different length and time scales. Furthermore, we show in which sense waiting time phenomena can occur in pattern forming systems.
Applied Mathematics Colloquium
Abstract: Particle transport by coherent vortices in laminar and turbulent flows gives rise to complex Lagrangian dynamics whose understanding is critical for a number of areas of engineering practice. I will review our earlier work and present new results underscoring the complexity of transport phenomena in 3D vortex dominated flows and revealing some striking dynamical similarities in seemingly disparate cases and flow regimes. I will explore via numerical simulations transport in two steady, chaotically advected flows: 1) in the interior of vortex breakdown bubbles; and 2) in a closed container with exactly counter-rotating lids. I will show that when rich Lagrangian dynamics emerge in the interior of perturbed vortex breakdown bubbles the rate at which passive tracers exit the bubble is the fractal curve known as the devil's staircase distribution. For the second case, I will analyze the sedimentation of initially suspended inertial particles to show that the competition of gravity and inertial forces gives rise to a striking fractal sedimentation regime also governed by a devil's staircase distribution. A very similar devil's staircase distribution will also be shown to emerge in the transport of sediment grains in the form of bedload by the turbulent horseshoe vortex upstream of an obstacle mounted on a mobile bed. Unlike the laminar sedimentation case, however, the bedload transport dynamics will be shown to be multifractal due to the inherent intermittency of transport by coherent turbulent vortices. Finally, I will present a new unsteady Eulerian sediment transport model for computing the instantaneous bedload flux in turbulent flows that is inspired by Lagrangian ideas. The model predicts the emergence of dynamically rich and statistically meaningful bed-forms that resemble those observed in nature.
Brown University Graduate School
Dissertation Defense Information Seminar
PDE Seminar