Lefschetz Center for Dynamical Systems Seminar
Please Note Change of Time and Place for This Week Only |
Abstract: We consider a differential equation presented by Pontryagin in his ODE's book (1962) to explain the condition for Lyapunov stability of the equilibrium -- representing constant speed performance -- of the Watt centrifugal governor coupled with a steam engine in the model of Maxwell (1868) - Vishnegradskii (1877).
After a discussion of the relevance of the study of this model as a paradigm for automatic control, we determine the values of the parameters (3 in Pontryagin's Equation) where the Lyapunov stability fails and where successive Hopf bifurcations of increasing order appear.
A bifurcation diagram is presented as a qualitative synthesis of the results obtained.
Work done in collaborations with L. Mello and D. Braga.
Center for Computational Molecular Biology Lecture
Organismic and Evolutionary Biology, Harvard University, Cambridge, MA Faculty Search Candidate Center for Computational Molecular Biology | |
Abstract: Evolutionary biology and evolutionary computation share a long-standing interest in the topography of fitness landscapes: the projection of organismal fitness values over genotype sequence space.
Specifically, local peaks (i.e. genotypes whose single-mutant neighbors are all of lower fitness) have long been thought to pose a singular challenge to Darwinian evolution, which can only increase population fitness. I show first that multiple fitness peaks may be regarded as a limiting case of a more general phenomenon: the selective inaccessibility of one or more mutational trajectories to high-fitness genotypes. Both are caused by a particular form of mutational interaction. I will also show that because evolving populations cannot be regarded as occupying a single point on the fitness landscape, local peaks do not stall Darwinian evolution.
Next, because theoretical work on fitness landscapes lacks empirical focus, I characterize the fitness landscape for an enzyme responsible for heightened bacterial antibiotic resistance. This landscape is single-peaked but a very large fraction of all mutational trajectories to the high resistance allele are selectively inaccessible as a consequence of mutational tradeoffs between structural and functional attributes of the enzyme. Interestingly, a principled quantitative treatment of these effects raises more questions than it answers concerning enzyme action and evolution in vivo. Nevertheless, as such intramolecular pleiotropic effects appear generic among missense mutations, I conclude that replaying the protein tape of life may be surprisingly repetitive.
It remains to be seen whether intermolecular interactions will similarly constrain Darwinian evolution at larger scales of biological organization, a question with diverse implications such as for the nature of metabolic modularity and the evolutionary significance of genetic recombination. This will be the topic of my chalk talk.
The Future of Biostatistics
Brown University Center for Statistical Sciences Seminar
2006 Lecture Series
Seven lectures celebrating the 10th anniversary of
The Center for Statistical Sciences
Abstract: It is common in applied research to have large numbers of variables measured on a modest number of cases, with a variety of types of data (continuous, binary, ordinal categorical, nominal categorical, etc.) Longitudinal data and other clustered data structures are also common.
This talk will present various methods that have emerged as part of an effort to develop broadly applicable and flexible model-based imputation methods for high-dimensional data sets. Key ideas include handling missing continuously- scaled items using factor-analysis ideas to reduce the number of covariance parameters to be estimated in a multivariate normal model (Song and Belin 2004), using growth-curve models and factor-analysis ideas together for longitudinal continuously-scaled variables (Wang and Belin 2002); using a parameter-exended Metropolis-Hastings algorithm to sample the correlation matrix in a multivariate probit model in a manner that lends itself to extensions to several ordinal variables (Zhang, Boscardin, and Belin 2003; Boscardin, Zhang, and Belin 2004), and applying the parameter-extended Metropolis- Hastings idea to a multinomial probit model in a manner that lends itself to extensions to several nominal categorical variables (Zhang, Boscardin, and Belin 2005). Examples are offered to illustrate the methods, and simulation studies are used to explore statistical properties and to compare procedures with potential alternative approaches.
Sponsored by the Charles K. Colson Lectureship and
Publication Fund
Co-Sponsored by the Bruce M. Bigelow Class of 1955 Lecture Series
Brown University
Joint Materials/Solid Mechanics Seminar Series
University of Michigan, Ann Arbor, Michigan | |
Abstract: Numerous mechanisms can give rise to instabilities and vibrations in sliding systems. These can generally be characterized as either elastodynamic (e.g. 'brake squeal' and stick-slip or frictional vibrations) or thermoelastic ('TEI', known in the automative industry as 'hot judder') The time scales of these processes differ by several orders of magnitude, so it is usual to neglect coupling between them -- i.e. to neglect thermal effects in elastodynamic analyses and to use the quasi-static approximation in thermoelastic analyses. The possible coupling between these two phenomena has recently been considered in several extremely idealized systems. First, we consider a thermoelastodynamic layer sliding against a rigid plane and constrained to one-dimensional displacements normal to the plane. A linear perturbation analysis shows that although the coupling is extremely weak, it has a destabilizing effect on the natural elastodynamic vibration of the layer at arbitrarily low sliding speeds. A numerical solution of the transient equations below the quasi-static (TEI) critical speed shows that an initial disturbance grows exponentially until periods of separation develop, after which the system approaches asymptotically to a steady state involving periods of contact and separation alternating at the lowent natural freguency of the elastodynamic system. With increasing sliding speed, the proportion of the cycle spent in contact is reduced and the maximum contact pressure increases. It is important to note that neither a quasi-static thermoelastic analysis, nor an elastodynamic analysis neglecting thermal expansion would predict instability in this speed range. The proposed mechanism might also provide an explanation of reported experimental observations of vibrations normal to the contact interface during frictional sliding. If an elastodynamic half plane slides against a rigid plane surface, frictional instabilities can occur in the absence of thermal effects if the friction coefficient is sufficiently high. However, thermoelastodynamic instabilities occur for lower friction coefficients and for most sliding speeds, though the stability behaviour now becomes considerably more complex. We also report some initial results of a finite element analysis of the coupled stability problem, which permits the extension of the argument to more realistic and complex geometries. In the long term, it is hoped that these effects can be incorporated into commercial CAE software.
Center for Computational Molecular Biology Lecture
University of California, San Diego Faculty Candidate Center for Computational Molecular Biology | |
Abstract: Variation in human DNA sequences account for a significant amount of the genetic risk factors for common disease such as hypertension, diabetes, Alzheimer's disease, and cancer. Identifying the common variation that influences susceptibility to disease will usher in a new era of personalized medicine where treatment decisions are based not only on clinical observations, but also take into account an individual's genetic makeup. Recent technological advances in high-throughput genotyping technology allow us for the first time to collect human variation information on a large enough scale to identify the variation involved in disease. This talk focuses on two challenges associated with the analysis of high-throughput genotype data. Since the new cost effective technologies obtain human variation information from both pairs of human chromosomes simultaneously, the first step in analysis of these datasets is computational prediction of the human variation on each chromosome or the haplotype phasing problem. We discuss the results of our collaboration with Perlegen Sciences on the phasing of whole genome human haplotypes. A second challenge is the association of whole genome variation data to phenotypic data or clinical traits. Using the inbred mouse as a model organism, we demonstrate how our methods are able to discover many regions in the mouse genome associated with phenotypes and how many of our predictions are consistent with genes known to influence specific traits.
Bio: Eleazar Eskin received his Ph.D. in Computer Science at Columbia University in October 2002. After graduation, he was a post-doctoral researcher at Hebrew University in Jerusalem, Israel. He is currently an Assistant Professor in Residence in the Department of Computer Science at the University of California, San Diego and affiliated with the California Institute of Telecommunications and Information Technology (Calit2).
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Brown Analysis Seminar
Abstract: If X is a compact set in Cn, R(X) denotes the uniform closure on X of all rational functions P/Q, with Q non-vanishing on X. The set X is called "rationally convex" if for every y in CnX there exists a polynomial Q with Q(y) = 0 and Q never 0 on X. It is known that if R(X) = C(X), then X is rationally convex.
Question: Let X be a compact rationally convex set on the unit sphere in C2. When does R(X) = C(X)?
We observe that X is analogous to a flat subset of a convex surface in R3 (e.g. a line segment on a circular cylinder), and we introduce two measures of "holomorphic flatness" of such an X. In terms of these notions, we give two sufficient conditions on the set X for R(X) = C(X).
Scientific Computing Seminar
Department of Aerospace Engineering, University of Michigan | |
Abstract: Discontinuous Galerkin methods are the Finite Element analyst's answer to Finite Volume methods. Originally inspired by upwind (Godunov-type) methods for the advection equation and hyperbolic systems, the DG community soon turned to the diffusion equation, with much less success. It seems that the DG approach is fundamentally unsuited for second-order operators. The most successful method of today, the Local Discontinuous Galerkin method of Shu and Cockburn requires that the diffusion equation be rewritten as a system of first-order equations. While working with first-order systems is computationally advantageous, and a general trend in CFD, it evades the question how to directly discretize a second-order operator.
In this lecture I will first show there are essential differences between discretizing the advection and diffusion equations: what works for one does not work for the other, and vice versa. This means that, when formulating a DG method for diffusion, one can not blindly copy what's done for the advection equation.
I will show, however, there is absolutely no conflict between the DG approach and the diffusion equation. In order to make it work two insights are needed:
(1) the realization that there are multiple representations of the numerical solution which all are equivalent in the weak sense, and that one may have to switch between these for the sake of getting useful schemes;
(2) for a second-order PDE integration by parts must be done TWICE in order to obtain the DG equations - which is not standard DG practice.
Next, I will present the Recovery method, developed from the above starting points. Specifically, a smooth locally recovered solution is used that in the weak sense is indistinguishable from the discontinuous discrete solution. The recovery principle creates schemes that are not included in the family of traditional DG diffusion schemes, and are potentially more accurate. A way is presented to extend the family so that some recovery-based schemes are included.
An eigenvalue/eigenvector analysis suggests that the order of accuracy of the recovery schemes may be as high as 2**(p+2}, i.e., exponential. This conclusion is supported by 1-D numerical tests, in which the new and old schemes are pitted against each other.
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Bram van Leer, Professor and Graduate Chair
Department of Aerospace Engineering
University of Michigan
Ann Arbor, MI 48109-2140
Tel: (734) 764-4305
Fax: (734) 763-0578
E-mail: bram@umich.edu
PDE Seminar
Department of Mathematics Colloquium
<--- 2006 Index