Lefschetz Center for Dynamical Systems Seminar
Abstract: More than a hundred years ago, C.A. Bjerknes and his son V.F.K. Bjerknes observed that two pulsating bubbles in a liquid will either attract or repel each other depending on whether their radial pulsations are in, or out, of phase. They also observed that the magnitude of the mutual force between the bubbles obeys an inverse square law. This hydrodynamic phenomenon has since come to be known as the secondary Bjerknes force. New equations of motion are presented which govern the translation and radial oscillations of two interacting bubbles. In the process of obtaining these equations, a new derivation is given of an elegant surface transport theorem from potential flow theory. A dynamical systems analysis of the equations of motion yields a two-body problem (classical mechanics) for the two bubble system. It is found that nonlinear motion of the bubbles can reverse the direction of the mutual force from that predicted by linear theory. The talk will conclude with a brief discussion of the parametric stability of a nonspherical bubble.
Brown University Center for Statistical Sciences Seminar
Abstract: Imaging is most helpful not only in enhancing the scientific understanding but also as undisputed aid to diagnosis of disorders. The evaluation of diagnostic imaging systems as well as pharmaceutical products used in imaging requires different statistical methodologies than therapeutic medical products. In the design phase of a diagnostic clinical trial, a great deal of attention needs to be paid to subject selection and to the reduction of statistical bias. At the analysis phase, different statistical tools including Receiver Operating Characteristic (ROC) methodology have proven to be quite useful in the evaluation of diagnostic imaging systems. The advantages of multiple reader studies are discussed.
Stochastic Systems Seminar
Abstract: The Brownian motion with small variance is probably the first example of random element in infinite dimensional space which is known to satisfy large deviation principle. It has been well established now that by using contraction principle (or some variance of the argument), the large deviation principle can be proved for quite general small perturbed diffusion processes with smooth coefficients. There are many interesting applications of how these results can be used to study difficult problems related to some singular perturbed PDE. It seems the story on the large deviations related to Brownian is complete. In this talk, we consider the one dimensional Brownian motion with small variance, its occupation time in the positive real line and the local time at the origin. We give the large deviation principle for this triple joint processes. The large deviation rate function is also given. We shall show some small perturbed diffusions with discontinuous coefficients which our results give an easy approach to establish their large deviation principle. We shall also show that the large deviation rate function given by Boue-Dupuis-Ellis for such processes is closely connected with the large deviation rate function of this triple joint processes.
PDE Seminar
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Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: The classical model of visual processing in cortex is a hierarchy of increasingly sophisticated representations, extending the model of simple to complex cells of Hubel and Wiesel. Relatively little quantitative modeling has been done in the last 15 years to explore the biological feasibility of this class of models to explain higher level visual processing, such as object recognition. I will review experimental results in viewpoint-invariant object recognition and describe here a new hierarchical model --- developed with Max Riesenhuber --- that seems to represent a possible strategy for solving this complex visual task, is consistent with several recent physiological experiments in inferotemporal cortex and makes testable predictions. A key element of the model is a softmax response function of some neurons where the strongest afferent determines the unit's output. The softmax operation was suggested by trying to find the computational equivalent in cortex of a scanning operation which is a key module in a family of successful computer vision algorithms that we have developed during the last few years.
Special Joint Seminar
Lefschetz Center for Dynamical Systems and
The Center for Fluid Mechanics
Abstract: In engineering computations of high Reynolds number turbulent flows, the large scales are simulated directly, but due to the broad range of length scales encountered in such flows, turbulence at the small scales typically needs to be modeled. Therefore, it is necessary to understand small scale turbulence and develop the corresponding statistical models.
In this talk, I will present recent work on understanding and modeling of small scale turbulence which focuses on: (1) The Rankine-Burgers vortex model, which can be used to address the anisotropic scalings and asymptotic scalings of the velocity structure function. (2) The mapping closure method, which is a systematic approach for calaulating the probability density function. These results provide new physical insights into small scale turbulence and will be used to improve current statistical models.
Brown University Center for Statistical Sciences Seminar
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Abstract: The accelerated failure time model is an appealing semi-parametric model which is a useful alternative to the Cox proportional hazards model in survival analysis. This model is a simple log-linear model. In this talk, first we review some existing methods for inferences about the regression coefficients. Almost all the methods in the literature are rather complex numerically. I will propose a simple and reliable method to analyze censored data under this model. The proposed method can be easily implemented using linear programming techniques with existing software for estimating the regression coefficients of the standard linear model based on the L1 norm. The new procedure is illustrated by analyzing a HIV -1 RNA data set from a study conducted by the AIDS Clinical Trial Groups. Generalization to the case with repeated RNA measurements will also be discussed with this example.
PDE Seminar
Brown University Center for Statistical Sciences Seminar
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Abstract: We consider the analysis of data from a national panel study on changes in the treatment practices of outpatient methadone treatment units. The analysis of this data set is challenging due to several difficulties: multiple longitudinal outcomes; non-ignorable non-responses; and missing covariates. Specifically, several variables were used to measure the effectiveness of methadone treatment practices for each unit. A substantial number of units did not respond (33%) during the follow-up. These dropout units tended to be units with less effective treatment practices and were hence non-ignorable. Finally, for the units that dropped out, their time-varying covariates were missing at the time of dropout. We propose a latent variable model for multivariate longitudinal outcomes, where the observed outcomes are related to a latent variable (e.g., treatment practices effectiveness), and the latent variable is associated with covariates through a linear mixed model. A selection model is then developed to model non-ignorable dropouts, where the dropout probability depends on the latent variable. To accommodate missing time-varying covariates at the time of dropout, a transition model for these covariates is proposed. Maximum likelihood estimates are obtained using the EM algorithm. We also investigate the asymptotic bias of parameter estimates when missing time-varying covariates are filled in using a naive approach such as last observation carried forward.
Department of Mathematics Colloquium
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