Special Division of Applied Mathematics Colloquium
Abstract: If a dynamical system with k, k>1, stable attractors, say equilibriums, is perturbed by a noise of intensity e<<1, then for (almost) any initial point x and time scale T(e)~exp{d/e}, under some natural assumptions, a unique equilibrium M(x,d) exists such that the perturbed trajectory, during the time interval (0,T(e)), spends most of the time near M(x,d). State M(x,d) is called meta- stable state (for the initial point x and time scale T(e)~exp{d/e}).
If the characteristics of the small noise are changing slowly, the metastable state also changes. In particular, if the changes are periodic, the system performs (in a sense) periodic oscillations. This effect is called stochastic resonance. The large deviation theory allows to give rigorous explanation to all these effects.
In the case of dynamical system which is close to Hamiltonian one, under certain conditions, a distribution on the set of attractors serves as the metastable state. This is a result of an interplay between averaging and large deviations.
Brown University Center for Statistical Sciences Seminar Series
Department of Quantitative Health Sciences, Cleveland Clinic | |
(Refreshments beginning at 3:15 p.m.) |
Abstract: It is customary to approach inference for high-throughput genomic data through the use of hypothesis testing. Another approach, however, is to adopt a variable selection paradigm, which offers some unique advantages. I discuss such an approach using what are called "spike and slab models". Some background on variable selection will be given to motivate this approach and several examples using software we have developed will be detailed.
Brown University-
Joint Materials/Solid Mechanics Seminar Series
Abstract: We have produced a stretchable form of silicon that consists of sub-micrometer single crystal elements structured into shapes with microscale periodic, wave-like geometries (Science, v 311, pp 208-212, 2006). When supported by an elastomeric substrate, this wavy silicon can be reversibly stretched and compressed to large strains without damaging the silicon. The amplitudes and periods of the waves change to accommodate these deformations, thereby avoiding significant strains in the silicon itself. Dielectrics, patterns of dopants, electrodes and other elements directly integrated with the silicon yield fully formed, high performance wavy metal oxide semiconductor field effect transistors, pn diodes and other devices for electronic circuits that can be stretched or compressed to similarly large levels of strain. There are many mechanics problems in stretchable electronics, and a few will be discussed in this talk.
Stochastic Systems Seminar
School of Operations Research and Industrial Engineering | |
Abstract: We consider an infinite-source Poisson process to model end user inputs to a data network. We assume that the sources initiate connections according to a Poisson process and that transmission rates and durations are independent random variables. We analyze the traffic process that is obtained by discretizing time in slots of length $\delta$ and consider the quantity of transmitted data in adjacent time intervals. We study this discrete time process as the slot length $\delta$ goes to $0$. This analysis extends and complements our earlier work on a related model in which we assumed independence of the transmission rates and the file sizes. It is striking that the two cases show rather different behaviour. When file size and rate are independent, cumulative input per slot converges marginally to a normal distribution, but in the model considered here we have an approximating distribution which is stable with infinite second moment. We also study dependence across time slots, characterize its slow rate of decay, and provide a detailed comparison of the two models.
Joint work with Bernardo D'Auria
Special Talk -- Stochastic Systems Seminar
Abstract: In this talk, I will discuss higher boundary regularity for the parabolic equations. A sharp condition for the solution to have kth order continuous derivatives with respect to both space and time variables near singular boundary points is obtained, for integers k\geq 2. The approach here is probabilistic. We don't know at present whether it is possible to give a proof only in terms of the theory of partial differential equations. The main difficulty is the absence of the analogues for the Burkholder-Davis-Gundy inequality, which plays an important role in the proofs.
Brown University-
Joint Materials/Solid Mechanics Seminar Series
Technion, 32000 Haifa, Israel merittel@technion.ac.i: www.technion.ac.il/~merittel | |
Abstract: Assessing the dynamic mechanical properties of structural materials is an important stage of the design process of structures that must withstand various impacts. This data is mostly needed for numerical simulation purposes, when the dynamic response and eventual failure of a structure is to be evaluated.
In this talk, we will address one specific dynamic failure mechanism, namely adiabatic shear banding. Adiabatic shear banding (ASB) is a catastrophic failure mechanism that may develop in certain ductile materials subjected to dynamic loading. The phenomenon itself consists of a narrow band of sheared material, in which the local temperature may reach a significant fraction of the melting temperature, as a result of thermomechanical coupling effects.
As of today, there is an overwhelming disparity between analytical-numerical models related to ASB formation, and experimental evidence aimed at verifying a specific criterion or simply bringing physical evidence. This excludes of course the wealth of information related to microstructural aspects of ASB.
The Dynamic Failure Laboratory at Technion has been investigating ASB formation from an experimental point of view. This talk will present new results on ASB formation in metals.
Three specific issues will be addressed, namely:
1. A physical criterion for the onset of ASB formation
2. The influence of hydrostatic pressure on ASB formation
3. The influence of geometrical imperfections on ASB formation
Brown University Center for Statistical Sciences Seminar
Arnold School of Public Health, University of South Carolina | |
Abstract: The development and application of diagnostics for cluster detection in spatio-temporal (ST) disease incidence data is considered. In addition we develop the use of local likelihood in the formulation of the Bayesian hierarchical model for the ST variation of the disease incidence. The diagnostics developed are post hoc in that they operate in the output from posterior samples (or in some cases on estimators that are non-Bayesian). The measures include both single region and neighborhood diagnostics, such a exceedence probability, lag risk difference, and cusum measures. Different measures tend to highlight different features of the clustering behavior whether its persistent spatial clusters or ST clusters. We also present results for a comparison of a standard random effect Bayesian model (SREST) and a local likelihood model developed for ST applications. In the local likelihood formulation a lasso parameter defining the size of a cylinder in ST is used to define a 'local' contribution to the likelihood. These local contributions are correlated, but the correlation is modeled at a higher level of the hierarchy. This leads to a relatively fast algorithm for cluster detection, but keeps within the Bayesian model paradigm. A comparison is made between these models in a simulation study to assess the behavior of different diagnostics.
Background Reference:
Hossain, M. and Lawson, A. (2006) Cluster Detection diagnostics
for small area health data.
Statistics in Medicine, 25, 771-786
Scientific Computing Seminar
Center for Science and Informatics (CWI) The Netherlands | |
Abstract: In my talk I will present geometric multigrid (mg) analysis for higher-order discontinuous Galerkin (DG) dicretization of elliptic problems. I will show that the mg-iteration scheme obtains optimal efficency if the smoother damps the discontinuities in the iteration error before it is restricted to the coarser meshes.
I will further introduce the embedded boundary discretization technique. This technique is of interest for elliptic boundary value problems defined on complex domains, if it is preferable to keep the mesh orthogonal. I show results of a singularly perturbed embedded boundary value problem that is solved with a fourth-order DG discretization and multigrid iteration.
PDE Seminar
Unharmonic Oscillators | |
Abstract: Many dispersive nonlinear Hamiltonian systems (such as NLS or nonlinear Klein-Gordon equations) are known to have solitary wave solutions. These solitary waves can be stable and moreover asymptotically stable. Further, one expects that in a rather general situation the long time asymptotics for any finite energy solution could be described as a superposition of solitary waves (plus dispersion).
We consider the U(1)-invariant Klein-Gordon equation in one dimension, coupled to finitely many anharmonic oscillators. We prove that in this model the weak global attractor is formed by the solitary waves (if the distances between the oscillators are sufficiently small).
Physically, the global attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. The main analytical tools are the Paley-Wiener arguments and the Titchmarsh Convolution Theorem.
This is a joint work with Alexander Komech, University of Vienna.
<--- 2006 Index