Abstract: In the context of random walk in a random environment (RWRE), large deviation principles (LDPs) are classified according to two criteria: (i) which random variable they are concerned with, and (ii) whether it is under the quenched measure (i.e., when the environment $\omega$ is frozen) or the averaged measure (i.e., when $\omega$ is integrated out). The most commonly studied random variable is the mean velocity of the walk. Such LDPs are called level-1. Level-2 refers to the empirical measures for the environment (from the point of view of the particle) and k-future steps of the walk (for some fixed $k\geq 0$). Finally, if all future steps are recorded, then the corresponding LDPs are known as level-3 (or process level). The so-called contraction principle lets us derive (and give variational formulas for the rate functions of) lower level LDPs from higher level ones. In some cases, we know that these variational formulas have unique minimizers which correspond to Markov processes whose kernels are obtained from the original RWRE kernel by a generalization of the h-transform technique of J.L. Doob, involving a tilting by harmonic functions. In this talk, I will give a survey of the cases where this connection between large deviations and h-transform has been established. The results for dimensions $d=1$, $d=2,3$ and $d\geq 4$ are rather different in flavor.
Center for Vision Research
Abstract: The ability to choose rapidly among multiple targets embedded in a complex perceptual environment is key to survival. Targets will differ in their reward value as well as in their low-level perceptual properties (e.g., visual saliency), how does the brain combine these two variables in visual search, and how does this compare with the optimal strategy? I will present psychophysical experiments and computational models suggesting that decisions, even in rapid visual search, are influenced by both target value and feature-contrast in a way that is consistent with an ideal Bayesian observer maximizing reward.
Center for Fluid Mechanics, Division of Applied Mathematics Fluids, Thermal and Chemical Processes Group, School of Engineering Joint Seminar Series
Abstract: The deposition of aqueous drops on non-wetting surfaces is an important canonical problem for many applications, including suppressing splash or rebound of sprayed herbicides on intrinsically hydrophobic plant leaves. The addition of a small amount of high molecular weight polymer has been demonstrated to suppress drop rebound on smooth hydrophobic surfaces. The high extensional viscosity of polymer solutions and the increased viscous dissipation near the receding contact line are cited as two distinct anti-rebound mechanisms. Using drop impact experiments on micro- and nano-textured surfaces in addition to smooth hydrophobic surfaces with controlled wetting characteristics, we examine the roles of viscosity, elasticity and inertia on expansion, retraction, and rebound of well-characterized viscoelastic fluids. We show that when a viscoelastic drop impacts a textured hydrophobic surface, the answer to the question "to bounce or not to bounce" depends critically on both the fluid viscoelasticity and the surface characteristics, including details of texture and its wetting characteristics. By adopting a stickslip flow model on textured surfaces with various topographic length scales and solid area fractions, we rationalize the dynamics leading to complete rebound following drop impact on nanotextured surfaces even for viscoelastic fluids.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: In this talk I will discuss two recent projects. First I will present a massively data driven approach to super-resolution. In contrast to recent image enhancement literature, we show that image databases can be more useful than internal image statistics if the right representations are used. Our super-resolution method can insert scene-specific textures and transitions beyond the capabilities of prior work. Second, I will introduce the new SUN Attribute Database. We use crowd-sourcing to discover a taxonomy of attributes and label 14,000 images from 700 scene categories with more than one hundred attributes. We show the first results for attribute recognition and argue that attributes are a natural way to describe the space of scenes.
Abstract: The primary research focus of the Howlett laboratory is the eukaryotic DNA damage response. Specifically, we study the molecular etiology of the rare chromosome instability disease Fanconi anemia (FA), using biochemical, cytogenetic, and genomic approaches. FA is clinically characterized by congenital defects, progressive pediatric bone marrow failure, and pronounced cancer susceptibility. At thecellular level FA is characterized by chromosome instability and hypersensitivity to DNA crosslinking agents. To date 15 FA genes have been identified, the protein products of which function cooperatively in the FA-BRCA pathway to repair DNA damage. Importantly, somatic and epigenetic inactivation of the FA-BRCA pathway is a frequent occurrence in cancer and bone marrow failure in the general (non-FA) population. Therefore the study of this rare disease stands to impart a greater understanding of the molecular origins of abnormal hematopoiesis and cancer susceptibility in general.
Abstract: BEACON, A Center for the Study of Evolution in Action, at Michigan State University provides an opportunity for evolutionary biologists and computer scientists studying evolutionary computation to exchange ideas and techniques from their parallel universes. While many of views of evolution are very similar between the two groups, the views of epistasis stand out as particularly distant from one another. In this talk I will outline a general mathematical model for the structure of a fitness landscape used in evolutionary computation and discuss the strengths and weaknesses for this model with respect to the needs of biologists and the data they have available. I will discuss some preliminary work on repackaging algorithms from evolutionary computation for use with biological data. I will give some examples on biological data.
Scientific Computing Seminar
Shallow-water equations with a non-flat bottom topography have been widely
used to model flows in rivers and coastal areas. An important difficulty
arising in these simulations is the appearance of dry areas, and standard
numerical methods may fail in the presence of these areas. These equations
also have steady-state solutions in which the flux gradients are non-zero
but exactly balanced by the source term.
In this presentation, we propose high-order discontinuous Galerkin and weighted essentially non-oscillatory methods, which can preserve the steady-state exactly, and at the same time are positivity preserving without loss of mass conservation. Some numerical tests are performed to verify the positivity, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.
Abstract: We will discuss some work on conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville type theorems, Harnack inequalities, and Bocher type theorems.
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