Center for Statistical Sciences Seminar
Abstract: This talk uses exponential-family random graph models (presented earlier in this seminar series) to examine the generative processes that give rise to systematic patterns in adolescent friendship networks. The methods incorporate both traditional demographic measures on individuals (age, sex, and race) and network measures for structural processes operating on individual, dyadic, and triadic levels. After a brief review of the methods, we apply them to the adolescent friendship networks for fifty-nine school groups from the National Longitudinal study of Adolescent Health (Add Health). We model friendship formation as a selection process constrained by individuals' sociality (propensity to make friends), selective mixing in dyads (friendships within race, grade, or sex categories are more likely), and closure in triads (a friends' friends are more likely to become friends), given local population composition. White and Blacks are the most consistently cohesive racial categories, while Hispanics present more variable patterns across schools at both the dyadic and triadic level. Grades are always highly cohesive, while females have greater triad closure than males given all other processes. The talk will conclude with a discussion of testing network model fit, and how network analysis may contribute to our understanding of sociodemographic processes.
Lefschetz Center for Dynamical Systems Seminar
Abstract: Parabolic problems with discontinuous nonlinearities arise in the description of many phenomena in the applied sciences, for instance: chemical reactor theory, porous medium combustion, and game theory (best response dynamics). Problems with nonlocal conditions appear in the modeling of diffusion and heat conduction, thermoelasticity, etc. we shall consider a parabolic problem with multivalued right hand side with integral initial condition. Several existence results will be presented.
Probability Seminar
Abstract: In this talk we present a recent result on the smoothness of the density for the solution of a semilinear heat equation with multiplicative space-time Gaussian white noise. We assume that the coefficients are smooth and the diffusion coefficient is not identically zero at the initial time. The proof of this result is based on the techniques of the Malliavin calculus, and the existence of negative moments for the solution of a linear heat equation with multiplicative space-time white noise.
Brown University Center for Statistical Sciences Seminar
Abstract: This tutorial will present an introduction to applying exponential-family random graph (ERG) models to network data using the statnet suite of R packages. After a very brief introduction to R, we will explore some of the basic functions in statnet to import, query, manipulate, visualize, and describe social network data. We will then move into fitting ERG models, simulating new networks from model fits, and assessing model degeneracy and model fit. Knowledge of R is helpful but not essential; participants may choose to follow along on their own laptops if they wish.
Applied Mathematics Colloquium
Abstract: Insects, like birds and fish, locamote via interactions between fluids and flapping wings and fins. Their motion is governed by the Navier-Stokes equation coupled to moving boundaries. In this talk, I will first describe how dragonflies fly. I will then describe the computational tools that we use to solve these problems, including a new immersed interface method to simulate multiple (rigid or flexible) wing interactions. I will show the computational results on the aerodynamics, and efficiency of flapping flight, and discuss questions related to animal locomotion: 1) Is insect flight optimal? 2) How does the efficiency of flappy flight compare to classical fixed-wing flight? 3) How might aerodynamic effects have influenced the evolution of insect flight?
Brown University Center for Statistical Sciences Seminar
Abstract: In biomarker validation studies, efficient use of information can lead to great savings of time and costs. In this project, our goal is to validate a risk prediction marker in various populations when the marker's classification accuracy as characterized by the ROC curve is invariant across populations. A default strategy is to study the marker's performance in different populations separately. Here we propose alternative methods that incorporate the constant ROC curve assumption into estimation. We illustrate this methodology in a real dataset by evaluating PCA3 as a risk prediction marker for prostate cancer among subjects with or without initial biopsy.
Center for Computational Molecular Biology Seminar
Abstract: Pattern discovery is a ubiquitous problem in many disciplines. It is especially prominent in recent years due to our greatly improved data-generation capabilities in science and technologies. The method I present here is motivated by the "motif-finding" and "module-finding" problems in biology, i.e., to find sequence patterns (i.e., "words") that seem to appear more frequent than usual in a given set of text sequences (i.e., sentences) and to find which of these "words" tend to co-occur in a sentence. A challenge in the motif-finding problem is that there are no spacings and punctuations between the words and the dictionary of "words" is unknown to us. Existing methods are mostly "bottom-up" approaches, i.e., to build up the dictionary starting with single-letter words and then concatenate some existing words that appear to occur next to each other in sentences more frequently than chance. Our new approach is a top-down strategy, which uses a tree structure to represent the relationship among all possible existing words and uses the EM algorithm to estimate the usage frequency of each word. It automatically trims down most of the incorrect "words" by letting their usage frequencies converge to zero. The module-finding problem is closely related to the well-known "market basket" problem, in which one attempts to mine association rules among the items in a supermarket based on customers' transaction records. It is also related to the two-way clustering problem. In this problem, we assume that the words are given, and our goal is to find subsets of words that tend to co-occur in a sentence. We call the set of co-occurring words (not necessarily orderly) a "theme" or a "module". We can generalize the dictionary model to the "theme"-model and use a similar EM-strategy to infer these themes. I will demonstrate its applications in a few examples including an analysis of Chinese medicine prescriptions and an analysis of a Chinese novel. This is based on a joint work with Ke Deng and Zhi Geng.
Lefschetz Center for Dynamical Systems Seminar
Abstract: Suppose that D denotes the open unit disc in the complex plane and f:D-->D is analytic map which has no fixed points in D. The classical Denjoy-Wolff theorem assserts that there exists w in the boundary of D such that, for all z in D, f^k(z) approaches w as k approaches infinity. (Here f^k denotes the composition of f with itself k times). Much later work of Beardon made clear that a crucial point is that f is nonexpansive with respect to an "appropriate" metric, in this case, the Poincare metric. If G is a bounded, open convex subset of a finite dimensional vector space V, one can define Hilbert's projective metric d on G and consider maps f:G-->G which are nonexpansive with respect to d. Many such maps arise in applications, e.g., "reproduction-decimation operators" from the theory of diffusion on fractals. We shall describe Denjoy-Wolff type theorems for such maps.
***Dissertation Defense Information***
Scientific Computing Seminar
Abstract: In this talk, we will present a unified framework for the construction of Schwarz Preconditioners for all the discontinuous Galerkin~(DG) approximations of elliptic problems that have been proposed up to the date. We shall introduce two level additive and multiplicative iterative methods for both symmetric and non-symmetric DG schemes, and we will discuss some new interesting features arising from the analyses, which have no analogue in the conforming case. Optimalrates of convergence will be established for these methods. Extensive numerical experiments will be presented to assess and support the theoretical results and to further illustrate the performance and robustness of the proposed preconditioners. It is based on joint work with P.F.Antonietti.
<--- 2008 Index