Center for Fluid Mechanics, Division of Applied Mathematics Fluids, Thermal and Chemical Processes Group, School of Engineering Joint Seminar Series
Abstract: Active particle suspensions, of which a bath of swimming bacteria is a paradigmatic example, are characterized by complex dynamics involving strong fluctuations and large-scale correlated motions. These motions, which result from the many-body interactions between particles, are biologically relevant as they impact mean particle transport, mixing and diffusion, with possible consequences for nutrient uptake and the spreading of bacterial infections. To analyze these effects, a kinetic theory is presented and applied to elucidate the dynamics and pattern formation arising from mean-field interactions. Based on this model, the stability of both aligned and isotropic suspensions is investigated. In isotropic suspensions, a new instability for the active particle stress is found to exist, in which shear stresses are eigenmodes and grow exponentially at low wavenumbers, resulting in large-scale fluctuations in suspensions of rear-actuated swimmers, or pushers, when the product of the linear system size with the suspension volume fraction exceeds a given threshold; no such instability is predicted for headactuated swimmers, or pullers. Numerical simulations of the kinetic equations are also performed, and applied to study the long-time nonlinear dynamics, which are characterized by transient particle clusters that form and break up in time, as well as complex chaotic flows correlated on the system size. The predictions from the kinetic model are also tested using direct numerical simulations based on a slender-body model for hydrodynamically interacting self-propelled particles. These simulations confirm the existence of a transition to large-scale correlated motions in suspensions of pushers above a critical volume fraction and system size, which is seen most clearly in particle velocity and passive tracer statistics. We also find that the collective dynamics of pushers result in giant number fluctuations, local alignment of swimmers, and strongly mixing flows. Extensions of this work to model chemotactic interactions with an external oxygen field as well as steric interactions in concentrated suspensions are also discussed.
Pattern Theory and Vision Seminar
Abstract: For naturally occurring data, the dimension of the given input space is often very large while the data themselves have a low intrinsic dimensionality. Spectral kernel methods are non-linear techniques for transforming data into a coordinate system that efficiently reveals the underlying structure -- in particular, the "connectivity" -- of the data. In this talk, I will focus on one particular technique -- diffusion maps -- but the analysis can be used for other spectral methods as well. I will give examples of various applications of the method in high-dimensional regression and density estimation. I will also present a new extension of the diffusion framework to comparing distributions in high-dimensional spaces with an application to image retrieval and texture discrimination. (Part of this work is joint with R.R. Coifman, D. Liu, C. Schafer and L. Wasserman.)
BU/Brown PDE Seminars
Abstract: In this talk, we will present an overview of recent theoretical, numerical and exper- imental work concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one- and higher-dimensional settings within Bose-Einstein condensates at the coldest temperatures in the universe (i.e., at the nanoKelvin scale). We will discuss how this ultracold quantum mechanical setting can be approximated at a mean-field level by a deterministic PDE of the nonlin- ear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons and vortices. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting future directions within this field.
BU/Brown PDE Seminars
Abstract: We will consider the second initial boundary problem in narrow domains of width $\varepsilon << 1$ for linear second order differential equations with nonlinear boundary con- ditions. Using probabilistic methods we show that the solution of such a problem converges as $\varepsilon \downarrow 0$ to the solution of a standard reaction-diffusion equation in a domain of reduced dimension. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front. In addition, an important and interesting prob- lem, which is related to the previous one, is the Wiener process with instantaneous reection in a narrow tube which, in contrast to before, is assumed to be non-smooth asymptotically. In this case, the limiting process is a Markov process that can be described by its generator. I will report these results as well as recent results on large deviations for such limiting process and how they can be used to study front propagation for reaction-diffusion equations in non-smooth narrow domains.
CCMB Seminars
Abstract: Dosage compensation is an important model system for defining the mechanisms of coordinate gene regulation because all of the genes on a single chromosome are specifically identified and co-regulated. The Male-Specific Lethal (MSL) complex is the key regulator of Drosophila dosage compensation because it increases transcript levels from active genes on the single male X chromosome to equalize gene dosage with females who have two copies of each X chromosome (Belote). Both cis-acting DNA sequences called MSL Recognition Elements (MREs) (Ref) and X-linked roX (RNA on X) non-coding RNA components (Ref) have been implicated in distinguishing the X chromosome from autosomes. However, the way in which the MSL complex specifically targets MRE sequences on the male X chromosome remained elusive because MREs are only two-fold X-enriched and known MSL components are insufficient for direct recognition of MREs (Ref). We recently identified the CLAMP (Coupling Lethal Adaptor for MSL Proteins) zinc-finger protein as one of many candidate MSL-regulators (Larschan et al., submitted). Here, we demonstrate that CLAMP serves as a critical link between MSL complex and MREs by directly interacting with MREs and targeting MSL complex to its high affinity sites. Furthermore, CLAMP and MSL complex exhibit an inter-dependent binding interaction that strongly increases occupancy of both factors at MSL complex high affinity sites. Even in the absence of MSL complex, CLAMP is highly enriched at potential ‘seed’ sites distributed along the length of the male X chromosome. Therefore, we propose the following novel mechanism for X chromosome recognition: 1) CLAMP directly recognizes MRE elements and is enriched at seed sites including the roX loci that serve as initial targets during X-identification; 2) A CLAMP-MSL inter-dependent association at high affinity sites concentrates MSL complex at these seed sites to generate X-specificity from a two-fold X-enrichment of MRE sequences. In this way, we provide key insight into how a single chromosome can be specifically recognized within a complex eukaryotic genome.
Computational Seminar
Abstract:
Particle methods are widely used in numerical approximations of systems because they can
provide a realistic approach that allows the approximate description of an evolving measure.
Recently it has become clear that by stepping outside the Monte-Carlo paradigm these methods
can be of higher order with effective and transparent error bounds while preserving many of the
other advantages (the maximum principal etc.).
A weakness of particle methods (particularly in the higher order case) is the tendency for the
number of particles to explode if the process is iterated and accuracy preserved. By introducing
patching and dynamic recombination into such methods one can retain the high order accuracy
and take advantage of the intrinsic smoothing property of many parabolic systems while
simplifying the support of the intermediate measures used in the iteration. The methods are
seriously heavy in computational terms per step but can blow away traditional adaptive methods
where accurate solutions are required. The methods have been applied to PDE and Filtering
problems with considerable success.
Joint work with Nicolas Victoir, Christian Litterer, Wonjung Lee.
Applied Math Colloquium
Abstract: How can one describe a probability measure of paths? And how should one approximate to this measure so as to capture the effect of this randomly evolving system. Markovian measures were efficiently described by Stroock and Varadhan through the Martingale problem. But there are many measures on paths that are not Markovian and a new tool, the expected signature provides a systematic ways of describing such measures in terms of their effects. We explain how to calculate this expected signature I the case of the measure on paths corresponding to a Brownian motion started at a point x in the open set and run until it leaves the same set. A completely new (at least to the speaker) PDE is needed to characterise this expected signature. Joint work with Ni Hao.
Scientific Computing Seminar
Abstract: We consider numerical methods for solving a class of PDE's arising in mathematical biology. The methods use a priori knowledge of the transport used to represent physiological states of individuals within the system to obtain their superconvergence properties. We discuss some specific biological systems that motivated the method development.
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