Colloquia 2003-2004
Michel Ianoz, École Polytechnique Fédérale de Lausanne, July 23, 2004
11:00 a.m., Stratton Hall Room 203
Title: Field-to-Transmission Line Coupling--Computational Models and Validation
Abstract: Scientists and engineers have faced the problem of computing voltages or currents induced by an electromagnetic field on a line or circuit for the first time in the 50's, when they tried to determine the electromagnetic effects of lightning on overhead power lines. A few years later an extension of the telegrapher's equation for the case of an external field illuminating a line was proposed and applied to calculations pertaining to Nuclear EMP effects on transmission lines. The lecture will introduce different models for field-to-transmission lines coupling calculations as a well as solutions of these coupling equations using specific Green functions and a matrix form called the BLT equation. Solutions to take into account the radiation of the lines at higher frequencies and more general configurations consisting of several lines with various loads will be discussed. The validation of the models will be presented using measurements performed on EMP simulators.
Hirokazu Ninomiya, Ryukoku University, June 22, 2004
11:00 a.m., Stratton Hall Room 203
Title: Asymptotic behavior of the solutions of Allen-Cahn equation in the two dimensional space
Abstract: In this talk we consider the dynamics of solutions of the Allen-Cahn equation in the two dimensional space. We assume that the nonlinear term is like a cubic function and that there exist two stable constant equilibria. We show the existence and the global stability of the traveling wave solutions whose contours look like "V-shape". I will also explain the asymptotic behavior of solutions.
Umberto Mosco, University of Rome, May 11, 2004
3:00 p.m., Stratton Hall Room 202
Title: Some variational problems with irregularities
Abstract: Smoothness, in the sense of classical PDEs, is lost in many mathematical models of interest in applications. Certain metric and variational notions - rooted in harmonic analysis and asymptotic homogenization theory - as well as some probability techniques originated in the theory of stochastic processes, come then to help. They allow us to study various kind of irregular structures, both discrete and continuous, which exhibit an intrinsic non-Euclidean behaviour. We present a brief overview of recent methods and results of this kind, by referring, in particular, to research perspectives in boundary value problems and optimal control problems.
Dimitri Zenkov, North Carolina State University, April 30, 2004
11:00 a.m., Stratton Hall Room 203
Title: Variational Integrators for Constrained Mechanical Systems
Abstract: Variational integrators proved to be very effective in long-term numerical simulations of unconstrained mechanical problems. Recently, such integrators were adopted for constrained mechanical systems. The talk will give an overview of this theory. Conservation of momentum by the numerical algorithm will be discussed.
Ralph Showalter, Oregon State University, April 27, 2004
11:00 a.m., Stratton Hall Room 203
Title: The interface of a poroelastic medium with viscous fluid or elastic solid
Abstract: We develop a model for the filtration of a single phase slightly compressible viscous fluid through a saturated poroelastic medium coupled to the slow flow in an adjacent open channel or in contact with a rigid or elastic solid.
James O'Malley, Department of Health Care Policy, Harvard Medical School, April 23, 2004
11:00 a.m., Stratton Hall Room 203
Title: Accounting for Treatment-Noncompliance and Missing Outcomes in Randomized Trials: Sensitivity to Model Assumptions
Abstract: Randomization is the medium through which causal inferences are drawn about the effect of treatment. When randomized trials are broken via noncompliance and non-response, analyses must account for these mechanisms to ensure that reliable results are obtained. Frangakis and Rubin (1999) recently developed a methods-of-moments (MOM) estimator to account for non-compliance and non-response in trials involving cross-sectional outcomes. The Frangakis and Rubin methodology assumes that a patient has an underlying latent trait that defines their compliance to the treatment regime under each possible treatment assignment. If outcomes are missing-at-random within compliance groups, the MOM estimator yields consistent estimates of the causal effect of treatment assignment. In this talk, we develop a maximum likelihood (ML) counterpart to the MOM estimator. We demonstrate that ML yields more efficient estimates on both normal and non-normal data, and that ML is surprisingly robust to assumptions made in order to identify the causal effect of treatment assignment. The methods will be applied to clinical trial data in Psychiatry, a field where non-compliance with treatment protocol is the norm rather than the exception.
Padmanabhan Aravind, Department of Physics, WPI, April 16, 2004
11:00 a.m., Stratton Hall Room 203
Title: Polytopes and Quantum Physics
Abstract: A polytope is a generalization of a polyhedron to four or more dimensions (with a hypercube perhaps being the best known example). This talk will trace the unusual role that polytopes have played in proofs of Bell's theorem in the last decade, drawing principally on the results of my own work. Bell's theorem, proved by John Bell in 1966, is a basic theorem concerning the interpretation of quantum mechanics. Following a brief introduction to Bell's theorem, I will show how the geometry of the 24-cell and the associated Reye's configuration provide an intriguing new proof of this theorem. The presentation will be aimed at an audience familiar with neither polytopes nor quantum physics and will try to illuminate this somewhat bizarre connection between 19th century mathematics and 20th century physics. I will close with some reflections on what interest all this may hold for a 21st century audience.
Luis Roman, Department of Mathematical Sciences, WPI, April 9, 2004
11:00 a.m., Stratton Hall Room 203
Title: A Measure of Managers Performance Based on the 'Stochastic Discount Factor'
Abstract:Hedge funds have been described as skill-based investment strategies. Skill-based strategies obtain returns from the unique skill or strategy of the trader. Because hedge funds are actively managed, trader skill is certainly important, as are the basic trading strategies behind most hedge funds investments. For these reasons, measuring the investment performance of fund managers remains an important research problem. A central goal of performance evaluation is to identify those managers who posses investment information or skill superior to that of the general investing public. However, the question of whether mutual fund managers can deliver expected returns in excess of naive benchmarks has long been controversial. The idea of a "Stochastic Discount Factor" (SDF)has been the ground of most of the performance measures studied until now, but, there are doubts and discrepancies among researchers who use the SDF as a measure of performance. In this talk, we present, briefly, some of the mathematical concepts underlying the stochastic discount factor (SDF), and attempt to clarify the ambiguity surrounding it. Second, we define a new measure of managers performance assuming that the dynamics of the underlying assets are described by a system of stochastic differential equations. These equations and assumptions are the same ones adopted in the classical theory of option pricing and other research areas of financial mathematics. In this framework, not only the existence and uniqueness (in case of complete markets) is granted, but, also there is a specific and innate form for the SDF. (Joint with Kathryn Wilkens, Department of Management, WPI)
Stilian Stoev, Boston University, April 2, 2004
11:00 a.m., Stratton Hall Room 203
Title: Simulation methods for linear fractional stable motion and FARIMA using the Fast Fourier Transform (Stilian Stoev and Murad S. Taqqu)
Abstract: We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric alpha-stable (S alpha S) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with S alpha S innovations}. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks.
We generate paths of the LFSM process by using Riemann-sum approximations of its S alpha S stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error.
We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustrate our method. We include MATLAB implementations of these simulation algorithms and indicate how the practitioner can use them.
Noah Kraut, March 26, 2004
11:00 a.m., Stratton Hall Room 203
Title: A Computational Solution to the Brownian First Passage Time Problem Using Girsanov's Theorem
Abstract: The Brownian first passage time problem is to compute the distribution of the first time the Brownian motion crosses a boundary. We will present a solution to the first passage time problem based on the Cameron-Martin transformation of path space. Althought the solution is obvious from a theoretical standpoint, surprisingly it leads to a simple numerical technique for computing the distributions associated to continuous boundaries. The computational method is rather general and we will demonstrate some applications including: how to price a Barrier option in the Black-Scholes model and how to embed the Brownian motion in an arbitrary distribution on the positive real numbers.
George Jumper,Air Force Research Laboratory, Hanscom AFB., MA, March 19, 2004
11:00 a.m., Stratton Hall Room 203
Title: High Altitude Optical and Clear Air Turbulence, Measurements, Models and Forecasts
Abstract: The Airborne Laser motivated AFRL interest in optical turbulence in the upper atmosphere. The VS Battlespace Environment Division has been involved in the measurement of optical turbulence for many years. More recently, we have been working on techniques to forecast the phenomena. Atmospheric buoyancy waves or "gravity waves" are suspected to be the primary mechanism for upper atmospheric turbulence. Gravity waves are also suspected to be the cause of high altitude clear air turbulence incidents. Dr. Jumper will present the methods used to measure and forecast turbulence. He will also describe a recent experimental campaign in France to attempt to correlate optical turbulence intensity to gravity wave strength, and he will present some preliminary results of that campaign.
Frederic Dias, Ecole Normale Superieure de Cachan, France, March 12, 2004
11:00 a.m., Stratton Hall Room 203
Title: What is new in the theory of water waves?
Abstract: The present talk focuses on classical problems in the theory of water waves. What do we mean by the classical problem of water waves? We mean the problem consisting in solving the Euler equations in a domain bounded above by a free surface (the interface between air and water) and below by a solid boundary (the bottom). The bottom can be at any depth. The driving force is due to gravity. The effects of surface tension might be equally important and can be included in the analysis. What makes the water-wave problem so difficult is not its governing equation which is linear (Laplace's equation), but its two nonlinear boundary conditions on the free surface. For a lot of coastal engineering applications, solutions given by the linearized water-wave problem are accurate enough, but for a number of practical applications the fully nonlinear problem must be solved. Moreover, the water-wave problem has attracted mathematicians for almost a century because of its extremely rich structure.
This talk is devoted to recent analytical, numerical and experimental results in the theory of water waves.
Jon-Lark Kim, University of Nebraska-Lincoln, February 17, 2004
11:00 a.m., Stratton Hall Room 203
Title: Low-Density Parity-Check Codes from Graphs and Finite Geometries
Abstract: A low-density parity-check (LDPC) code is a binary linear code with a sparse parity check matrix H (i.e., a few 1s in H) such that any two distinct rows of H have at most one 1 in common. The first construction of LDPC codes is random, due to Gallager (1963). After forgotten for about 20 years, LDPC codes were revisited by Tanner who introduced a graphic representation of LDPC codes, called the Tanner graphs. MacKay and Neal (1996) demonstrated that LDPC codes approach the Shannon limit under the iterative decoding known as the sum-product algorithm. Hence LDPC codes have become very interesting both practically and mathematically. It has been one of main questions in coding theory today to construct LDPC codes combinatorially/algebraically to beat random LDPC codes. In this talk we give two new approaches to construct LDPC codes, one from bipartite graphs and the other from finite geometries.
Constantin Bacuta, Pennsylvania State University, February 13, 2004
11:00 a.m., Stratton Hall Room 203
Title: Applications of the Multilevel Theory to Regularity Estimates for PDE
Abstract: The multilevel representation of Sobolev norms on bounded domains can be viewed as the Fourier representation of Sobolev norms on the whole space. Embedding relation between various interpolation spaces on bounded domains can be efficiently analyzed using tools from multilevel (multigrid) theory. As a consequence, regularity estimates for elliptic boundary value problems on polygonal domains in terms of Sobolev and Besov norms are proved. New sharp finite element error estimates are deduced.
Gregory Panasenko, Saint Etienne University and Pennsylvania State University, February 12, 2004
2:00 p.m., Stratton Hall Room 202
Title: Method of Partial Asymptotic Decomposition of Domain
Abstract: The method of partial asymptotic decomposition of domain (Mathematical Models and Methods in Applied Sciences, v.8, no 1 (1998), pp.139-156) reduces the dimension of the problem or simplifies the problem in some other way in the main part of the domain keeping the initial formulation in the remaining part and prescribing the asymptotically precise conditions on the interface. This method is applied to problems of the structural mechanics and to the mechanics of composites and it allows a multi-scale modelling combining the macroscopic and microscopic descriptions in the same model.
Ivan Blank, University of Louisville, February 6, 2004
11:00 a.m., Stratton Hall Room 203
Title: Compactness, Regularity, and Asymptotics in Some Overdetermined Problems
Abstract: We show a method to eliminate a type of mixed asymptotics in certain free boundary problems, and we give two examples of its application. We first employ this technique to attack a situation which arises in inverse problems, and then we turn to a composite membrane problem involving an eigenvalue optimization. The method involves rescuing a blowup proceedure from a problem of concentration in a case of borderline regularity. Because of the lack of regularity, it appears that these problems cannot be tackled by the monotonicity formula of Alt, Caffarelli, and Friedman (1984), or by the more recent monotonicity formula of Caffarelli, Jerison, and Kenig (2002).
Darko Volkov, New Jersey Institute of Technology, February 3, 2004
11:00 a.m., Stratton Hall Room 203
Title: Integral equation methods for the statics and the dynamics of an electrified fluid bridge
Abstract: The experimental and theoretical study of the statics and the dynamics of electrified fluids is a topic of great interest for applied mathematicians and engineers. Controlling the dynamics of fluids by subjecting them to electric fields is an important tool in a variety of micro technology type applications. For example, the production of very high quality sprays of structured droplets or particles such that each particle is made of a core of a certain substance surrounded by another one, is of particular importance for encapsulation of food additives, targeted drug delivery, and special material encapsulation processing.
Integral equation methods have been successfully used to study the velocity field of liquid bridges collapsing under surface tensions. I will discuss in my talk how we used an integral equation method for determining the electric field between two plates set at a di erent potential and connected by a dielectric liquid bridge. An important part of our work consisted of finding a way of calculating in a fast and very accurate fashion the adapted Green’s function to this problem. I will present a numerical algorithm that determines equilibrium shapes (or lack of), reached when surface tensions and electric stress balance out, starting from an initial shape.
Finally, I will discuss preliminary results for the dynamical problem where both the electric field and the velocity field have to be solved for. We are particularly interested in numerically simulating how electric fields can be instrumental in controlling the dynamics of pinching and breaking up of liquid bridges or jets, and in studying the singularity that appears in finite time during that process. I will also outline directions for future research such as the very interesting case of strong fields and the formation of the so called Taylor cones. These are conical singularities that develop as the shape of a droplet becomes more and more elongated. They are involved in a new technique for the production of micrometer/nanometer size droplets.
Mihai Sirbu, Carnegie Mellon University, January 30, 2004
11:00 a.m., Stratton Hall Room 203
Title: A Two-Person Game for Pricing Convertible Bonds
Abstract: A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to keep the bond, thereby collecting coupons, or to convert it to stock. The bondholder wishes to choose a conversion strategy to maximize the bond value. Subject to some restrictions, a convertible bond can be called by the issuing firm, which presumably acts to maximize equity value and thus to minimize the bond value. This creates a two-person game, and we model the bond price as the value of this game. We show, however, that under our standing assumption (dividends are paid at a lower rate than the money market rate) this game reduces to one of two optimal stopping problems, and the relevant stopping problem can be determined a priori, i.e., without first solving the convertible bond pricing problem.
Because of dividend payments, the partial differential equation describing the pricing function becomes nonlinear. This means that our analysis involves a fixed point problem. We also prove that for large time to maturity the value of the convertible bond approaches the value of the perpetual convertible bond.
The presentation is based on joint work with Steven E. Shreve.
R. Lee DeVille, Rensselaer Polytechnic Institute, January 20, 2004
11:00 a.m., Stratton Hall Room 203
Title: Rigorous Multiple-timescale Analysis for ODE and PDE
Abstract: We will describe the methodology of normal forms and concentrate on applications to weakly nonlinear but singularly perturbed ODE and PDE. The applications that we consider will include weakly nonlinear ODE, linear beam equations, and the Klein-Gordon equation. Although various formal methods exist to understand these problems, we show that normal form techniques can be both rigorous and algorithmizable. Furthermore, in certain problems there are subtle limitations to multiscale analysis (e.g. one can progress only to a certain timescale and the formal analysis is no longer valid), and we will show how the normal form theory can signal these difficulties.
Micah Dembo, Boston University, December 12, 2003
11:00 a.m., Stratton Hall Room 202
Title: Field Theories of the Cytoplasm: Application to the Mechanics of Neutrophils
Abstract: Much experimental data exist on the mechanical properties of neutrophils, but so far, they have mostly been approached within the framework of liquid droplet models. This has two main drawbacks: (i) It treats the cytoplasm as a single phase when in reality, it is a composite of cytosol and cytoskeleton; (ii) It does not address the problem of active neutrophil deformation and force generation. To fill these lacunae, we develop here a comprehensive continuum-mechanical paradigm of the neutrophil that includes proper treatment of the membrane, cytosol, and cytoskeleton components. We further introduce a continuum formulation of active force production via cytoskeletal swelling and via a generalized surface polymerization mechanism.
These tools, permit quantitative computational analysis of three classic experiments: the passive aspiration of a neutrophil into a micropipette, the active extension of a pseudopod by a neutrophil exposed to a local stimulus, and the crawling of a neutrophil inside a micropipette toward a chemoattractant against a varying counterpressure. Principal results include: (i) Membrane cortical tension is a global property of the neutrophil that is affected by local area-increasing shape changes. We argue that there exists an area dilation viscosity caused by the work of unfurling membrane storing wrinkles and that this viscosity is responsible for much of the regulation of neutrophil deformation. (ii) If there is no swelling force of the cytoskeleton, then it must be endowed with a strong cohesive elasticity to prevent phase separation from the cytosol during vigorous suction into a capillary tube. (iii) We find that both swelling and polymerization force models are able to provide a unifying explanation of the combined experimental data from all three experiments. However, force production required in the polymerization model is beyond what is expected from a simple short-range Brownian ratchet model.
Jun Zhu, University of Wisconsin, December 5, 2003
2:00 p.m., Stratton Hall Room 304
Title: A Multiresolution Tree-Structured Spatial Linear Model
Abstract: Multiresolution spatial models are able to capture complex dependence correlation in spatial data. Because of the multiresolution structures, spatial process prediction can be obtained by direct and fast computation algorithms. However the existing multiresolution models usually assume a simple constant mean structure, which may not be suitable in practice. In this presentation, we focus on a multiresolution tree-structured spatial model developed by Huang et al. (2002) and extend the model to incorporate a linear regression mean. An Expectation-Maximization algorithm is adopted to obtain the maximum likelihood estimates of the model parameters and the corresponding information matrix. Given the estimated parameters, a one-pass change-of-resolution Kalman-filter algorithm is implemented to obtain the best linear unbiased predictor of the true underlying spatial process. For illustration, the methodology is applied to map optimally crop yield in a Wisconsin field, after accounting for the field conditions by a linear regression.
Phil Everson, Swarthmore College, December 2, 2003
3:00 p.m., Stratton Hall Room 203
Title: Bayesian Hierarchical Models for American Football Scores
Abstract: The distribution of points scored in recent National Football League (NFL) games is roughly bell-shaped, but clearly not Normal. In American football points are accumulated primarily in increments of 3 or 7 (for field-goals and touchdowns) and occasionally 2, 6 or 8 (for safeties, touchdowns with a failed extra-point attempt, and touchdowns with a 2-point conversion). I will describe two models for football scores: one is based on a bivariate Normal approximation with a fix-up for the discreteness; the other assumes Poisson distributions for the numbers of touchdowns and field-goals scored by each team. In each model the mean parameters for a game are assumed to be linear combinations of parameters specific to the home-team and the road-team involved. A level-2 model for the team parameters connects the outcomes for all games. It is possible to simulate outcomes for future games by using a Markov Chain Monte Carlo algorithm to generate parameter values from their posterior distribution given the games already played, then following the model for points scored given the teams involved. I will show how the models performed during the current NFL season and use them to make predictions for upcoming games.
Gabriele Nebe, Abteilung Reine Mathematik, Universitaet Ulm, November 20, 2003
11:00 a.m., Stratton Hall Room 202
Title: Lattices and sperical designs
Abstract: A lattice is the set of all integral linear combinations of a basis in a euclidean vector space. A measure for the error correcting properties of a lattice is its density, that is the maximal density of a packing of equal spheres that are centered around the lattice points. The densest lattices are known up to dimension 8.
I present a recent construction of dense lattices that uses spherical designs:
The lattice vectors of minimal length form a finite set $X$ of points on a sphere. If X is a spherical 4-design (which means that the mean value over X equals the integral over the sphere for all polynomials of degree less than or equal to 4), then the lattice is called strongly perfect. Strongly perfect lattices are local maxima of the density function. They can be classified in small dimensions. In higher dimensions one uses representation theory of finite groups and also modular forms to construct strongly perfect lattices.
Nicole A. Lazar, Carnegie Mellon University, November 14, 2003
11:00 a.m., Stratton Hall Room 202
Title: Warping, Combining, and Jackknifing Brains: Statistical Issues in the Creation of fMRI Group Maps
Abstract: Psychologists who work with functional magnetic resonance imaging (fMRI) often wish to make statements about the functional brain behavior of a group of subjects. In order to do this, it is necessary to create a so-called "group map," which combines the activation of the individual subject brains in a (statistically) sensible fashion. Creating these maps involves defining a test statistic at each voxel of the brain that takes into account the level of activity displayed by every subject at that voxel, and thresholding to reveal areas of group activation. In this talk I show how, using some simple (and some not so simple) techniques for combining information already in use in the statistics literature, it is possible to generate group maps of the brain from the fMRI data of individuals. I also describe a "leave one out" approach for assessing the sensitivity of combining techniques to the effects of individual subjects. Finally, I will argue that the desiderata for a good combining technique in fMRI can be different from those that are generally assumed in other data analysis contexts.
Triantaphyllos Akylas, MIT, November 7, 2003
11:00 a.m., Stratton Hall Room 202
Title: Nonlinear Internal Gravity Wave Beams
Abstract: Recent numerical simulations and field observations reveal that thunderstorms often give rise to gravity-wave disturbances that propagate in the atmosphere along specific directions depending on frequency. These wave-beam structures are akin to the arms of the classical "St Andrews-Cross" wave pattern (see, for example, the front cover of the paperback version of the text "Waves in Fluids" by M.J. Lighthill) due to a localized source oscillating at a frequency below the buoyancy frequency in a uniformly stratified, inviscid Boussinesq fluid. An asymptotic theory for the propagation of modulated two-dimensional and axisymmetric nonlinear wave beams will be presented, that takes into account viscous effects as well as refraction effects due to the presence of a mean flow and nonuniform buoyancy frequency. The theory explains why a linear approach has been useful in interpreting certain observations of isolated beams in the atmosphere. On the other hand, nonlinear effects play an important part in the reflection of wave beams from a slope and in collisions of obliquely propagating beams. The theoretical predictions are consistent with numerical simulations and experiments.
Changfeng Gui, University of Connecticut, November 6, 2003
4:00 p.m., Stratton Hall Room 203
Title: On a conjecture of De Giorgi
Abstract: In 1979 De Giorgi conjectured that some entire solutions to the stationary Allen-Cahn equation, which is a well-known model in phase transition, must only depend on one direction if the dimension is less than 9. The conjecture is closed related to the Bernstein problem in geometry on the complete minimal graph surfaces in the entire space. In this talk, I will explain the connection of the conjecture with phase transition and minimal surfaces, discuss in some details the recent progresses.
Tomaz Pisanski, Colgate University and University of Ljubljana, November 6, 2003
11:00 a.m., Stratton Hall Room 202
Title: Generalized Petersen Graphs and Configurations
Abstract: Generalized Petersen graphs G(n,r) form an interesting family of trivalent graphs that has been studied in the past quite extensively. The family contains a number of very important graphs, such as G(5,2) - the Petersen graph, G(10,2), the skeleton of a dodecahedron, G(10,3), the Levi graph (incidence graph) of the renowned Desargues configuration. The purpose of this talk is to explore further links between geometric configurations and certain generalized Petersen graphs. We will briefly touch upon the theory of astral and polycyclic configurations and the theory of covering graphs. In particular, we will show that G(18,5) is the Levi graph of the unique smallest astral, triangle-free (v_3) configuration. We will also present a combinatorial argument why quadrangle-free astral (v_3) are hard to find. This is joint work in progress with Marko Boben, Branko Grunbaum and Arjana Zitnik.
Erin Terwilleger, University of Connecticut, October 31, 2003
11:00 a.m., Stratton Hall Room 202
Title: Third Order Commutator and Product BMO
Abstract: Let M_b denote the operation of multiplication by b. Let H_j denote the Hilbert transform performed in the j-th coordinate for j=1,2,3. Consider the third order commutator defined by [[[M_b,H_1],H_2],H_3]. In joint work with Michael Lacey, we show that the operator norm of the third order commutator on L^2(R^3) is comparable to the norm of b in the product space BMO(R x R x R) of S. Y. Chang and R. Fefferman. This fact has some well known equivalences. In particular, it allows us to deduce similar properties for Hankel operators. In addition, it gives the weak factorization result for the Hardy space H^1=H^2 x H^2 in the three parameter setting. The corresponding fact for the first order commutator [M_b,H] is classical. The corresponding fact for [[M_b,H_1],H_2] is a theorem of Ferguson and Lacey. In this talk, we follow the strategy of Ferguson and Lacey and find that a new ingredient is needed at a particular juncture of the argument.
Luis Roman, WPI, October 10, 2003
11:00 a.m., Stratton Hall Room 203
Title: What, Why, and Where Stochastic Differential Equations
Abstract: Physical phenomena of interest in science and technology are very often theoretically simulated by means of models which correspond to ordinary or partial differential equations. These equations are in general nonlinear and, as such, their solution is usually a difficult task. In addition, the more realistic mathematical models show a random character. For instance, many times some of the parameters and/or initial data are not known with complete certainty due to lack of information, uncertainty in the measurements or incomplete knowledge of the mechanisms themselves, and in practice most systems undergoes perturbations from the surrounding ambient. To compensate this lack of information and to have a more realistic description of the system one introduces noise in the equations. This results in "Stochastic Differential Equations". In this talk, I will give a brief introduction to this subject and present examples from different fields.
Samantha Cook, Harvard University, October 7, 2003
3:00 p.m., Stratton Hall Room 203
Title: Using historical control data to impute missing outcomes when clinical trials become open-label
Abstract: Fabrazyme is a drug being developed by Genzyme Corporation to treat Fabry disease, a rare disease caused by an enzyme deficiency. Fabrazyme is currently in Phase 4 trials with the US Food and Drug Administration (FDA), and due to positive preliminary results may become commercially available before the trial ends. If Fabrazyme does become available early, patients in the control group of the Phase 4 trial will have the opportunity to go off trial protocol and begin taking Fabrazyme, thus invalidating their control status after this point. The main outcome measured in the clinical trial is serum creatinine, which is generally accepted as a surrogate for more serious outcomes such as death or renal failure. We propose a method to impute "missing" controls' serum creatinine values, as if they had stayed on placebo. This involves fitting a Bayesian hierarchical changepoint model to data from historical controls, incorporating information learned from the historical controls into a similar model for randomized controls, and using this model and observed on-protocol data to impute serum creatinine values for patients randomized to control who started taking Fabrazyme early. Once missing serum creatinine values have been multiply imputed, the completed data sets can be analyzed as planned and their results combined in a straightforward way. This is joint work with Don Rubin and Elizabeth Stuart.
Florin Catrina, Worcester Polytechnic Institute, September 12, 2003
11:00 a.m., Stratton Hall Room 203
Title: Critical Exponents and Critical Dimensions
Abstract: We shall discuss the Euler-Lagrange equations for an energy functional related to weighted Hardy-Sobolev inequalities. The talk is concerned with the bifurcation of solutions for some problems in the unit ball and the notion of "critical dimension".
Joseph E. Flaherty, Rensselaer Polytechnic Institute, August 28, 2003
4:00 p.m., Stratton Hall Room 203
Title: Adaptive and Parallel Discontinuous Galerkin Methods for Hyperbolic Systems
Abstract: The discontinuous Galerkin (DG) method provides an appealing approach to address problems having discontinuities, such as those that arise in hyperbolic conservation laws. Originally developed for neutron transport problems and first analyzed by Le Saint and Raviart, the technique lay dormant for approximately fifteen years before becoming popular. It is now being used to solve ordinary differential equations and hyperbolic, parabolic, and elliptic partial differential equations.
The method may be regarded as cross between a finite volume and finite element method and it has many of the good properties of both. Thus, for example (i) it can sharply capture solution discontinuities relative to a computational mesh; (ii) it simplifies adaptation since inter-element continuity is neither required for h-refinement (mesh refinement and coarsening) nor p-refinement (method order variation); (iii) it conserves the appropriate physical quantities (e.g., mass, momentum, and energy) on an elemental basis; (iv) it can handle problems in complex geometries to high order; (v) regardless of order, it has a simple communication pattern to elements sharing a common face that simplifies parallel computation. With a discontinuous basis, however, the DG method produces more unknowns for a given order of accuracy than traditional finite element or finite volume methods and this may lead to some inefficiency. Limiting strategies to reduce spurious oscillations when high-order methods are applied to problems with discontinuities is difficult when DG methods are applied on unstructured meshes.
We describe several theoretical and computational aspects of the DG method as it applies to hyperbolic problems. Our focus is on fluid dynamics; however, the method is capable of handling virtually any other application. We develop the method using a local formulation of Cockburn and Shu and describe choices for bases, numerical flux functions, solution limiting and stabilization, shock and boundary layer detection, local time stepping, and a posteriori error estimation. Discontinuity detection reduces the need for limiting; thereby, retaining a high order of accuracy in regions where solutions are smooth. We further describe environments and data structures for serial and parallel adaptive computation, adaptive h- and p-refinement procedures on structured and unstructured meshes, and anisotropic mesh refinement. These techniques are illustrated for transient compressible flow problems.
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