Brown Analysis Seminar
Abstract: Two classical boundary value problems for the Laplacian are the Dirichlet and Neumann problems. A third type of boundary value problem is the mixed problem, where we specify Dirichlet data on a portion of the boundary and Neumann data on the remaining portion. In this talk I will discuss recent progress on the mixed problem in bounded Lipschitz domains. This is joint work with Russell Brown.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract:
With concerns of bioterrorism, the advent of new epidemics that spread with person-to-person contact, such as SARS, and the rapid growth of on-line social networking websites, there is currently great interest in building statistical models that emulate social networks. Stochastic network models can provide insight into social interactions and increase understanding of dynamic processes that evolve through society. A major challenge in developing any stochastic social network model is the fact that social connections tend to exhibit unique inherent dependencies. For example, they tend to show a lot of clustering and transitive behavior, heuristically described as ``a friend of a friend is a friend.'' It might be reasonable to expect that covariate similarities, or ``closeness'' in social space, should somehow be related to the probability of connection for some social network data. The relationship between covariates and relations is likely to be complex, however, and may in fact be different in different regions of the covariate space. Here, we present a new socio-spatial process model that smoothes the relationship between covariates and connections in a sample network using relatively few parameters, so the probabilities of connection for a population can be inferred and likely social network structures generated. Having a predictive social network model is an important step toward the exploration of disease transmission models that depend on an underlying social network.
[pizza will be provided]
Boston University/Brown University PDE Seminar
Abstract: Kinetic theory provides a coarse-grained alternative to the integrate-and-fire neuronal network description. In the limit of infinitely short conductance responses, a Boltzmann-type differential-difference equation can be derived for the probability density function of the neuronal voltage. A Fokker-Planck and a mean-field equation can be derived in the limit of small and vanishing conductance fluctuations, respectively. The talk will present detailed solutions to these equations, describing both the steady asynchronous and synchronously-oscillating states of the network, and will also discuss the effects of the network architecture. The mean-field provides exact solutions for the steady asynchronous state. For scale-free neuronal networks, it can be used to argue that the distributions of the firing rates and neuronal activity correlations are also scale free. The steady asynchronous state is also described by asymptotic solutions of the Fokker-Planck equation, using the size of the neuronal conductance fluctuations as the small parameter. in addition, the Fokker-Planck equation can also be used to describe the likelihood and temporal period of synchronous network oscillations, in which all the neurons fire in unison. The likelihood of synchrony is computed combinatorially using the network oscillation period and the voltage probability distribution. The oscillation period is found from a first-passage-time problem described by a Fokker-Planck equation, which is solved analyticaly via an eigenfunction expansion. The voltage probability distribution is found using a Central-Limit-Theorem-type argument via a calculation of the voltage cumulants. Differences between oscillations in all-to-all coupled and scale-free networks will also be discussed.
Boston University/Brown University PDE Seminar
Graduate Student Seminar
Abstract: From the fallout of my topics exam comes a much simpler analytic number theory talk attempting to explain what Selberg's eigenvalue conjecture is and how it corresponds to the generalized Ramanujan conjecture. I'm going to make a concerted effort to pitch this talk to everyone.
Scientific Computing Seminar
Abstract: The need to interpret and extract possible inferences from high-dimensional datasets has led over the past decades to the development of dimensionality reduction and data clustering techniques. Scientific and technological applications of clustering methodologies include among others computer imaging, data mining and bioinformatics. Current research in data clustering focuses on identifying and exploiting information on dataset geometry and on developing robust algorithms for noisy datasets. Recent approaches based on spectral graph theory have been devised to efficiently handle dataset geometries exhibiting a manifold structure, and fuzzy clustering methods have been developed that assign cluster membership probabilities to data that cannot be readily assigned to a specific cluster. In this talk, we develop a novel fuzzy spectral clustering algorithm that combines seamlessly the strengths of spectral approaches to clustering with various desirable properties of fuzzy methods. The developed methodology is applied to image segmentation and registration problems. Work in collaboration with Philip K. Maini, Radek Erban, and Ornella Cominetti of the University of Oxford.
PDE Seminar
Abstract: In this talk, I will discuss general models used to model flux of fluids in porous media: two and three phase isothermal flows, non-isothermal flows with phase change, combustion problems. These models have important applications, such as, oil recovery, improved oil recovery, soil and groundwater remediation, CO2 sequestration. I will present the PDEs and we I will discuss some analytic, semi-analytic and numerical techniques used for obtaining the solutions of these PDE. I will present also a brief discussion of open questions in different problems.