September 2007
The Real Zeros Property or An Example of Analysis helping Combinatorics and Discrete Probability
Friday, 9/14/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Miklos Bona (University of Florida)
ABSTRACT: Let a_0, a_1,... a_n be a sequence of non-negative real numbers. If the polynomial A(x)=\sum_{i=1}^n a_ix^n has real roots only, then that has a plethora of interesting combinatorial and probabilistic consequences for the numbers a_i. We will review some of these consequences. We will then turn to two specific examples, set partitions with big blocks, and Stirling permutations studied by Gessel and Stanley. We will show that both have the real zeros property when counted according to a natural parameter, the number of blocks and the number of descents, respectively. We then use these facts to show that the distribution of the number of blocks in the first case, and the number of descents in the second case both converge to a normal distribution.
For more information, e-mail ma-chair@wpi.edu.
Convergence of critical points of an approximating functional for brittle fracture in 1d
Friday, 9/21/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gilles Francfort (Université Paris-Nord)
ABSTRACT: In a one dimensional setting, critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.
For more information, e-mail ma-chair@wpi.edu.
Scaling, Self-similarity, and the Renormalization Group in Partial Differential Equations
Friday, 9/28/2007 3:30 PM-4:30 PM
Bartlett Center
Harold J. Gay Lecture Series
SPEAKER: Grigory Isaakovich Barenblatt, Professor in Residence, Department of Mathematics, University of California at Berkeley
ABSTRACT: Scaling (self-similar) solutions to the partial differential equations entered the applied mathematics field 200 years ago. Until recently they were treated mostly as "exact special" solutions to some very specific problems--elegant, sometimes useful for qualitative investigations of the models but, in general, very limited in their significance elements of the general theories.
Gradually, it was recognized that the value of these solutions is much more significant: they are the intermediate asymptotics to the solutions to wider classes of problems when the influence of the details of the initial and/or boundary conditions already disappeared, but the solution is still far from its ultimate form. The appearance of computers did not reduce but increased the value of the scaling solutions.
In some cases (in fact, such cases are rather rare) the scaling solutions can be obtained using the dimensional analysis. However, as a rule this is not the case: scaling solutions appear due to the invariance of the problem to an additional group (note,--group, not semigroup), which we identify as the renormalization group.
A survey of these topics will be presented in this lecture; illustrative examples will be used.
(Refreshments at 3:00 PM in Stratton Hall, Room 107)
For more information, e-mail ma-chair@wpi.edu.
October 2007
Non-classical solutions of the Buckley-Leverett equation in the context of two-phase flow
Friday, 10/26/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Bert Peletier (Leiden University)
ABSTRACT: We discuss an extension of the Buckley-Leverett (BL) equation describing two-phase flow in porous media. This extension includes a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases. We derive existence conditions for traveling wave solutions of the extended model. This leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. We obtain non-monotone weak solutions of the initial-boundary value problem for the BL equation consisting of constant states separated by shocks, confirming results obtained experimentally.
For more information, e-mail ma-chair@wpi.edu.
November 2007
Convergence of equilibria of thin elastic beams
Friday, 11/9/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Maria Giovanna Mora (SISSA, Trieste, Italy)
ABSTRACT: In a series of recent papers a hierarchy of lower dimensional theories for nonlinearly elastic thin beams has been rigorously derived, starting from three-dimensional elasticity, by means of Gamma-convergence.
This approach guarantees convergence of minimizers of the 3d elastic energy to minimizers of the reduced problem. In this talk we will discuss the convergence of (possibly non-minimizing) stationary points of the 3d elastic energy. These are joint results with S. Mueller and M. G. Schultz.
For more information, e-mail ma-chair@wpi.edu.
Entire Solutions and Their Symmetries
Friday, 11/16/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Changfeng Gui (University of Connecticut)
ABSTRACT: In this talk, I will discuss various symmetries associated with entire solutions of certain partial differential equations and systems. Entire solutions usually arise in blow-up procedure, and are important in understanding solutions in general, in particular the singularities of general solutions. The symmetry of solutions, such as radial symmetry, transitional symmetry and symmetries with respect to discrete group actions, are indeed characteristic to the type of singularities. A survey of recent results on symmetry properties related to phase transition will be given. Existence of symmetric quadruple junction solutions for multiple phase separation will also be presented.
For more information, e-mail ma-chair@wpi.edu.
Patient-Specific MRI-Based 3D FSI RV/LV/Patch Multi-Layer Anisotropic Models for Pulmonary Valve Replacement Surgery and Patch Optimization
Monday, 11/19/2007 12:00 PM-1:00 PM
Goddard Hall, 227
SPEAKER: Dalin Tang (Computational Mathematics and Biomedical Engineering, WPI)
ABSTRACT: A patient-specific right/left ventricle and patch (RV/LV/Patch) combination model with fluid-structure interactions (FSI) was introduced to evaluate and optimize human pulmonary valve replacement/insertion (PVR) surgical procedure and patch design. Cardiac Magnetic Resonance (CMR) imaging studies were performed to acquire ventricle geometry, flow velocity and flow rate for healthy volunteers and patients needing RV remodeling and PVR before and after scheduled surgeries. CMR-based RV/LV/Patch FSI models were constructed to perform mechanical analysis and provide accurate assessment for RV mechanical conditions and cardiac function. These models include a) fluid-structure interactions, b) isotropic and anisotropic material properties, c) two-layer construction with myocardial fiber orientation, and d) active contraction. Both pre- and post-operation CMR data were used to adjust and validate the model so that predicted RV volumes reached good agreement with CMR measurements (error < 2%). Two RV/LV/Patch models were made based on pre-operation data to evaluate and compare two PVR surgical procedures: i) conventional patch with little or no scar tissue trimming; ii) small patch with aggressive scar trimming and RV volume reduction. Our modeling results indicated that: a) patient-specific CMR-based computational modeling can provide accurate assessment of RV cardiac functions; b) PVR with a smaller patch and more aggressive scar removal led to reduced stress/strain conditions in the patch area and may lead to improved recovery of RV functions. More patient studies are needed to validate our findings.
Acknowledgement: This paper is a collaborative work. Contributions from co-authors are happily acknowledged: Prof. Chun Yang, WPI and Beijing Normal Univ.; Professors Tal Geva and Pedro J. del Nido, Harvard Medical School, Children's Hospital, Boston. This research was supported in part by NIHR01 HL63095 (PJdN), NIH-NHLBI 5P50 HL074734 (Clinical Trial, PI-Geva; Co-Investigator del Nido), and NSF-China Project Nonlinear PDEs in Geometry (10371001).
This is a joint talk with the WPI Biomedical Engineering Department.
For more information, e-mail ma-chair@wpi.edu.
On the d bar-Neumann Problem
Friday, 11/30/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Der-Chen Chang (Georgetown University)
ABSTRACT: In this talk, we shall discuss recent progress of the d bar-Neumann problem. We obtain solving operator for inhomogeneous Cauchy-Riemann equation. Then we discuss the "possible" optimal estimates of the solution.
For more information, e-mail ma-chair@wpi.edu.
December 2007
Alip Mohammed (York University) Topic TBA
Friday, 12/7/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alip Mohammed (York University)
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
January 2008
A Rigorous Treatment of Energy Extraction From A Rotating Black Hole
Friday, 1/18/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Joel Smoller (University of Michigan, Ann Arbor)
ABSTRACT: The Cauchy problem is considered for the scalar wave equation in the Kerr (rotating black hole) geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximum energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou's result for the Penrose process. The main mathematical tool is our previously derived integral representation of the propagator. The crucial ingredient in the proof is a time-independent energy estimate for the outgoing wave.
For more information, e-mail ma-chair@wpi.edu.
Remarks on non-local operator
Friday, 1/25/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Uemura Toshihiro (Kobe University of Commerce, Japan/ University of Connecticut)
ABSTRACT: Consider the nonlocal operator L having k(x,y) as a (Levy-type) kernel and the (symmetric) quadratic form E having k(x,y) as a (jumping) kernel.
In this talk, we will reveal a connection between L and E. That is, under a mild assumption for k(x,y), we can show that the quadratic form E is obtained by integrating the so-called Carre du Champ operator G relative to L. In the last, we will apply it to the case of stable-like processes.
If Lu is an alpha-order fractional Laplacian with a positive alpha which is at most 2, the norm E is the alpha order Sobolev norm of u and the corresponding stochastic process is a so-called symmetric alpha-stable process.
When alpha=2, the L is the Laplace operator, E is the Dirichlet integral and the process is nothing but a Brownian motion.
For more information, e-mail ma-chair@wpi.edu.
Applied Mathematics Research within DOEs Office of Advanced Scientific Computing Research
Friday, 1/25/2008 3:00 PM-4:00 PM
Stratton Hall, 202
SPEAKER: Homer Walker (WPI and U.S. Department of Energy)
ABSTRACT: The Office of Advanced Scientific Computing Research (ASCR), one of six interdisciplinary program offices within the Department of Energy's Office of Science, plays a central role in maintaining US leadership in scientific computation by supporting research in applied mathematics, computer science, and high-performance networks as well as by providing leading-edge computational and networking resources. In this presentation, I will discuss applied mathematics research within ASCR, covering the nature and scope of support for research and outlining some upcoming activities and future trends.
For more information, e-mail ma-chair@wpi.edu.
February 2008
Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature
Friday, 2/1/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Henrique Versieux (Courant Institute)
ABSTRACT: We study the numerical solution of a PDE describing the relaxation of a crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth order evolution equation. It can be seen as the radient flow of a convex but non-smooth energy with respect to the $H^{-1}$ inner product. Our numerical scheme uses implicit discretization in time and a mixed finite-element approximation in space. The singular character of the energy is handled using regularization, combined with a primal-dual method that remains robust as the regularization parameter tends to zero. We study the convergence of this scheme, both theoretically and numerically.
For more information, e-mail ma-chair@wpi.edu.
Boundary Layers of Incompressible Fluids with Navier Boundary Conditions
Friday, 2/15/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Yaguang Wang (Shanghai Jiaotong University & Northwestern University)
ABSTRACT:In this talk, we are interested in the small viscosity limit for the incompressible Navier-Stokes equations with the Navier friction boundary condition. First, by multi-scale analysis we formally study the asymptotic behavior of solutions when the viscosity goes to zero for several different cases of the slip length depending on the viscosity, and derive problems of boundary layers, secondly, we rigorously study the stability of boundary layers for the incompressible Navier-Stokes equations with anisotropic viscosities by using energy method, when the slip length is larger than the square root of the vertical viscosity, in which even though the boundary layer appears in the lower order of solutions but it yields the vorticity of fluids being unbounded in the viscosity limit.
For more information, e-mail ma-chair@wpi.edu.
Mathematics of Molecular and Cellular Biology
Friday, 2/29/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Roger Lui (WPI)
ABSTRACT: The study of molecular and cellular biology includes topics from DNA and RNA all the way to cell movement and chemotaxis. Currently, this is a very active branch of mathematical biology and many models are proposed in scientific journals every month. By and large, people don't know how to solve these problems because they are extremely difficult or because people don't know how to model them. In this talk, I shall illustrate these two points by discussing three examples taken from protein folding, biochemical network (cell signaling), and cell motility. I shall keep the technicality to a minimum.
For more information, e-mail ma-chair@wpi.edu.
March 2008
Portfolios and Risk Premia for the Long Run
Friday, 3/7/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Paolo Guasoni (Boston University)
ABSTRACT: This paper develops a method to derive optimal portfolios and risk-premia explicitly in a general diffusion model, for an investor with power utility and in the limit of a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk.
In models with several assets and a single state variable, long-run portfolios and risk-premia admit explicit formulas up the solution of an ordinary differential equation, which characterizes the principal eigenvector and its corresponding eigenvalue of a elliptic operator. Multiple state variables lead to a partial differential equation, which is solvable for most models of interest.
For each value of the relative risk aversion parameter, the paper derives the long-run portfolio, its implied risk-premia and pricing measure, and their performance on a finite horizon.
For more information, e-mail ma-chair@wpi.edu.
Levi L. Conant Lecture Series presents Jeffrey Weeks
Monday, 3/24/2008 3:00 PM-4:00 PM
Olin Hall, 107
SPEAKER: Jeffrey Weeks, Recipient of the 2007 AMS Levi L. Conant Prize
TITLE: The Shape of Space
ABSTRACT: When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a "multiconnected universe."
Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our universe.
The only prerequisites for this talk are curiosity and imagination. Middle school and high school students, people interested in astronomy, and all members of the WPI community are welcome to attend.
Coffee and tea will be available one half hour before the lecture.
For more information, e-mail ma-chair@wpi.edu.
Gelification and wax deposition in crude oils
Friday, 3/28/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Antonio Fasano (Departments of Mathematics U. Dini, University of Florence, Italy)
ABSTRACT: Waxy crude oils are mineral oils with a large content of heavy hydrocarbons(making what we call "wax"), normally dissolved. Various phenomena may occur when they are cooled down to sufficiently low temperatures: solid wax is segregated in the form of suspended crystals, the crystals may entangle trapping the oil and making a gel, the existence of thermal gradients may induce the migration of the still dissolved wax. This complex of phenomena may result in the formation of a solid deposit at the cold wall. The typical situation is the one of a submarine pipeline and the final consequence can be the pipe obstruction by the deposit.
A mathematical model will be presented describing the cooling process of a
static oil, combining all the previously mentioned effects. The model consists of a system of partial differential equations of various nature and contains several unknown interfaces. Existence of a solution is proved and qualitative properties are discussed.
For more information, e-mail ma-chair@wpi.edu.
Cardiac Systems Biology and Multi-Scale Modeling (Jointly with BE department)
Monday, 3/31/2008 12:00 PM-1:00 PM
Higgins Laboratories, 218
SPEAKER: Andrew McCulloch (BE Dept, UCSD)
ABSTRACT: Computational models of the heart can be integrative in several important ways. First, they permit information integration of genome scale data sets that would otherwise be difficult to interpret and understand. We illustrate this in the analysis of high-throughput phenotypic and metabonomic data on cardiac function in the fruitfly drosophila melanogaster. We use drosophila as a model organism for studying cardiac hypoxia tolerance, and integrating these data with genome-scale metabolic models. Systems models of cardiac myocytes achieve functional integration, by predicting how the functions of individual network components combine to give rise to physiological functions. We illustrate this with our recent models of -adrenergic regulation of myocyte excitation-contraction coupling. Multi-scale computational models aim to achieve structural integration across physical scales of biomedical organization from molecule to organize. We illustrate this with examples from our recent research on arrhythmia mechanisms in genetic disease and the effects of external interventions such as pacing and pericardiectomy on ventricular-vascular coupling in vivo.
For more information, e-mail ma-chair@wpi.edu.
April 2008
Coping with Uncertain Volatility when Pricing Equity Options
Friday, 4/4/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gunter Meyer (Georgia Institute of Technology)
ABSTRACT: In finance the volatility of an asset value is a measure of its randomness. It has the same influence on pricing options as the diffusivity has on the temperature in heat conduction. Volatility is not observable and we shall outline some attempts at estimating or bounding it from market data. Then we shall discuss the derivation and numerical solution of the nonlinear Black Scholes Barenblatt equation, a Partial-Integro-Differential equation, which yields upper and lower bounds on options prices for assets following a jump diffusion process with uncertain volatility. The basic tools are the maximum principle for the heat equation and a two point value problem solver based on the Riccati transformation.
For more information, e-mail ma-chair@wpi.edu.
Periodic nonlinear Schroedinger equation and photonic crystals
Friday, 4/11/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alexander Pankov (College of William and Mary)
ABSTRACT: We discuss the problem of existence of finite energy solutions to stationary periodic nonlinear Schroedinger equations. As an application we prove the existence of gap solitons in sertain models of one- and two-dimensional photonic crystals. (A photonic crystal is an artificial optical medium that has spatially periodic structure).
For more information, e-mail ma-chair@wpi.edu.
Math Awareness Week Lecture
Thursday, 4/17/2008 3:00 PM-4:00 PM
Higgins Laboratories, 218
SPEAKER: Tommy Ratliff, Wheaton College
TITLE: A Little Geometry, a Little Linear Algebra, and a Lot of Insight into Voting
ABSTRACT: The 2000 Presidential election taught many people that the winner of the popular vote may differ from the winner in the Electoral College. However, this is only a small example of the inconsistencies that are possible: If there are more than two candidates, then the voting procedure can have as much impact on the outcome as the preferences of the voters. We usually think of voting in the context of political elections, but the same framework used to understand elections can give insights into any group decision process.
We will see that a little geometry and linear algebra can go a long way toward explaining many voting paradoxes and how to avoid them.
Pizza & soda will be provided.
For more information, e-mail ma-chair@wpi.edu.
Creating materials with a desired refraction coefficient
Friday, 4/25/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alexander G. Ramm (Kansas State University)
ABSTRACT: Many-body scattering problem is solved asymptotically when the size of the particles tends to zero and the number of the particles tends to infinity.
A method is given for calculation of the number of small particles and their boundary impedances such that embedding of these particles in a bounded domain, filled with known material, results in creating a new material with a desired refraction coefficient.
The new material may be created so that it has negative refraction, that is, the group velocity in this material is directed opposite to the phase velocity.
Another possible application consists of creating the new material with some desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle. In this application it is assumed that the incident plane wave has a fixed frequency and a fixed incident direction.
An inverse scattering problem with scattering data given at a fixed wave number and at a fixed incident direction is formulated and solved.
For more information, e-mail ma-chair@wpi.edu.
May 2008
Adaptive Finite Element Discretization of a Phase Field Model of Brittle Fracture
Friday, 5/9/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Christoph Ortner (Oxford University)
ABSTRACT: Adaptive finite element methods are a popular tool for numerical analysts for the efficient discretization of PDEs with singular or "near singular" solutions. In recent years, significant progress has been made in their analysis, and we now have a good understanding of their convergence and optimality.
In this talk, I will begin by describing adaptive finite element methods and their analysis at a simple model problem and then show how these ideas can be used to efficiently discretize the Ambrosio--Tortorelli functional, a "phase-field model" of brittle fracture. If time permits, I will present a convergence proof for an adaptive optimization algorithm for this highly non-convex functional.
For more information, e-mail ma-chair@wpi.edu.
Yaxiang Yuan--CANCELED
Friday, 5/16/2008 11:00 AM-12:00 PM
Stratton Hall, 203
For more information, e-mail ma-chair@wpi.edu.
Quasistatic crack growth in elasto-plastic materials
Tuesday, 5/20/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gianni Dal Maso (SISSA, Trieste, Italy)
ABSTRACT: We present a variational model of quasistatic crack growth in elasto-plastic materials, which fits the general framework of the energy formulation of rate independent evolution problems developed by Mielke. The dissipation distance takes into account the effects of plastic flow and crack production. The model predicts that the crack tips belong to the closure of the plastic region.
For more information, e-mail ma-chair@wpi.edu.
September 2007
The Real Zeros Property or An Example of Analysis helping Combinatorics and Discrete Probability
Friday, 9/14/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Miklos Bona (University of Florida)
ABSTRACT: Let a_0, a_1,... a_n be a sequence of non-negative real numbers. If the polynomial A(x)=\sum_{i=1}^n a_ix^n has real roots only, then that has a plethora of interesting combinatorial and probabilistic consequences for the numbers a_i. We will review some of these consequences. We will then turn to two specific examples, set partitions with big blocks, and Stirling permutations studied by Gessel and Stanley. We will show that both have the real zeros property when counted according to a natural parameter, the number of blocks and the number of descents, respectively. We then use these facts to show that the distribution of the number of blocks in the first case, and the number of descents in the second case both converge to a normal distribution.
For more information, e-mail ma-chair@wpi.edu.
Convergence of critical points of an approximating functional for brittle fracture in 1d
Friday, 9/21/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gilles Francfort (Université Paris-Nord)
ABSTRACT: In a one dimensional setting, critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.
For more information, e-mail ma-chair@wpi.edu.
Scaling, Self-similarity, and the Renormalization Group in Partial Differential Equations
Friday, 9/28/2007 3:30 PM-4:30 PM
Bartlett Center
Harold J. Gay Lecture Series
SPEAKER: Grigory Isaakovich Barenblatt, Professor in Residence, Department of Mathematics, University of California at Berkeley
ABSTRACT: Scaling (self-similar) solutions to the partial differential equations entered the applied mathematics field 200 years ago. Until recently they were treated mostly as "exact special" solutions to some very specific problems--elegant, sometimes useful for qualitative investigations of the models but, in general, very limited in their significance elements of the general theories.
Gradually, it was recognized that the value of these solutions is much more significant: they are the intermediate asymptotics to the solutions to wider classes of problems when the influence of the details of the initial and/or boundary conditions already disappeared, but the solution is still far from its ultimate form. The appearance of computers did not reduce but increased the value of the scaling solutions.
In some cases (in fact, such cases are rather rare) the scaling solutions can be obtained using the dimensional analysis. However, as a rule this is not the case: scaling solutions appear due to the invariance of the problem to an additional group (note,--group, not semigroup), which we identify as the renormalization group.
A survey of these topics will be presented in this lecture; illustrative examples will be used.
(Refreshments at 3:00 PM in Stratton Hall, Room 107)
For more information, e-mail ma-chair@wpi.edu.
October 2007
Non-classical solutions of the Buckley-Leverett equation in the context of two-phase flow
Friday, 10/26/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Bert Peletier (Leiden University)
ABSTRACT: We discuss an extension of the Buckley-Leverett (BL) equation describing two-phase flow in porous media. This extension includes a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases. We derive existence conditions for traveling wave solutions of the extended model. This leads to admissible shocks for the original BL equation, which violate the Oleinik entropy condition and are therefore called nonclassical. We obtain non-monotone weak solutions of the initial-boundary value problem for the BL equation consisting of constant states separated by shocks, confirming results obtained experimentally.
For more information, e-mail ma-chair@wpi.edu.
November 2007
Convergence of equilibria of thin elastic beams
Friday, 11/9/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Maria Giovanna Mora (SISSA, Trieste, Italy)
ABSTRACT: In a series of recent papers a hierarchy of lower dimensional theories for nonlinearly elastic thin beams has been rigorously derived, starting from three-dimensional elasticity, by means of Gamma-convergence.
This approach guarantees convergence of minimizers of the 3d elastic energy to minimizers of the reduced problem. In this talk we will discuss the convergence of (possibly non-minimizing) stationary points of the 3d elastic energy. These are joint results with S. Mueller and M. G. Schultz.
For more information, e-mail ma-chair@wpi.edu.
Entire Solutions and Their Symmetries
Friday, 11/16/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Changfeng Gui (University of Connecticut)
ABSTRACT: In this talk, I will discuss various symmetries associated with entire solutions of certain partial differential equations and systems. Entire solutions usually arise in blow-up procedure, and are important in understanding solutions in general, in particular the singularities of general solutions. The symmetry of solutions, such as radial symmetry, transitional symmetry and symmetries with respect to discrete group actions, are indeed characteristic to the type of singularities. A survey of recent results on symmetry properties related to phase transition will be given. Existence of symmetric quadruple junction solutions for multiple phase separation will also be presented.
For more information, e-mail ma-chair@wpi.edu.
Patient-Specific MRI-Based 3D FSI RV/LV/Patch Multi-Layer Anisotropic Models for Pulmonary Valve Replacement Surgery and Patch Optimization
Monday, 11/19/2007 12:00 PM-1:00 PM
Goddard Hall, 227
SPEAKER: Dalin Tang (Computational Mathematics and Biomedical Engineering, WPI)
ABSTRACT: A patient-specific right/left ventricle and patch (RV/LV/Patch) combination model with fluid-structure interactions (FSI) was introduced to evaluate and optimize human pulmonary valve replacement/insertion (PVR) surgical procedure and patch design. Cardiac Magnetic Resonance (CMR) imaging studies were performed to acquire ventricle geometry, flow velocity and flow rate for healthy volunteers and patients needing RV remodeling and PVR before and after scheduled surgeries. CMR-based RV/LV/Patch FSI models were constructed to perform mechanical analysis and provide accurate assessment for RV mechanical conditions and cardiac function. These models include a) fluid-structure interactions, b) isotropic and anisotropic material properties, c) two-layer construction with myocardial fiber orientation, and d) active contraction. Both pre- and post-operation CMR data were used to adjust and validate the model so that predicted RV volumes reached good agreement with CMR measurements (error < 2%). Two RV/LV/Patch models were made based on pre-operation data to evaluate and compare two PVR surgical procedures: i) conventional patch with little or no scar tissue trimming; ii) small patch with aggressive scar trimming and RV volume reduction. Our modeling results indicated that: a) patient-specific CMR-based computational modeling can provide accurate assessment of RV cardiac functions; b) PVR with a smaller patch and more aggressive scar removal led to reduced stress/strain conditions in the patch area and may lead to improved recovery of RV functions. More patient studies are needed to validate our findings.
Acknowledgement: This paper is a collaborative work. Contributions from co-authors are happily acknowledged: Prof. Chun Yang, WPI and Beijing Normal Univ.; Professors Tal Geva and Pedro J. del Nido, Harvard Medical School, Children's Hospital, Boston. This research was supported in part by NIHR01 HL63095 (PJdN), NIH-NHLBI 5P50 HL074734 (Clinical Trial, PI-Geva; Co-Investigator del Nido), and NSF-China Project Nonlinear PDEs in Geometry (10371001).
This is a joint talk with the WPI Biomedical Engineering Department.
For more information, e-mail ma-chair@wpi.edu.
On the d bar-Neumann Problem
Friday, 11/30/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Der-Chen Chang (Georgetown University)
ABSTRACT: In this talk, we shall discuss recent progress of the d bar-Neumann problem. We obtain solving operator for inhomogeneous Cauchy-Riemann equation. Then we discuss the "possible" optimal estimates of the solution.
For more information, e-mail ma-chair@wpi.edu.
December 2007
Alip Mohammed (York University) Topic TBA
Friday, 12/7/2007 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alip Mohammed (York University)
ABSTRACT: TBA
For more information, e-mail ma-chair@wpi.edu.
January 2008
A Rigorous Treatment of Energy Extraction From A Rotating Black Hole
Friday, 1/18/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Joel Smoller (University of Michigan, Ann Arbor)
ABSTRACT: The Cauchy problem is considered for the scalar wave equation in the Kerr (rotating black hole) geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximum energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulou's result for the Penrose process. The main mathematical tool is our previously derived integral representation of the propagator. The crucial ingredient in the proof is a time-independent energy estimate for the outgoing wave.
For more information, e-mail ma-chair@wpi.edu.
Remarks on non-local operator
Friday, 1/25/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Uemura Toshihiro (Kobe University of Commerce, Japan/ University of Connecticut)
ABSTRACT: Consider the nonlocal operator L having k(x,y) as a (Levy-type) kernel and the (symmetric) quadratic form E having k(x,y) as a (jumping) kernel.
In this talk, we will reveal a connection between L and E. That is, under a mild assumption for k(x,y), we can show that the quadratic form E is obtained by integrating the so-called Carre du Champ operator G relative to L. In the last, we will apply it to the case of stable-like processes.
If Lu is an alpha-order fractional Laplacian with a positive alpha which is at most 2, the norm E is the alpha order Sobolev norm of u and the corresponding stochastic process is a so-called symmetric alpha-stable process.
When alpha=2, the L is the Laplace operator, E is the Dirichlet integral and the process is nothing but a Brownian motion.
For more information, e-mail ma-chair@wpi.edu.
Applied Mathematics Research within DOEs Office of Advanced Scientific Computing Research
Friday, 1/25/2008 3:00 PM-4:00 PM
Stratton Hall, 202
SPEAKER: Homer Walker (WPI and U.S. Department of Energy)
ABSTRACT: The Office of Advanced Scientific Computing Research (ASCR), one of six interdisciplinary program offices within the Department of Energy's Office of Science, plays a central role in maintaining US leadership in scientific computation by supporting research in applied mathematics, computer science, and high-performance networks as well as by providing leading-edge computational and networking resources. In this presentation, I will discuss applied mathematics research within ASCR, covering the nature and scope of support for research and outlining some upcoming activities and future trends.
For more information, e-mail ma-chair@wpi.edu.
February 2008
Numerical analysis of a steepest-descent PDE model for surface relaxation below the roughening temperature
Friday, 2/1/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Henrique Versieux (Courant Institute)
ABSTRACT: We study the numerical solution of a PDE describing the relaxation of a crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth order evolution equation. It can be seen as the radient flow of a convex but non-smooth energy with respect to the $H^{-1}$ inner product. Our numerical scheme uses implicit discretization in time and a mixed finite-element approximation in space. The singular character of the energy is handled using regularization, combined with a primal-dual method that remains robust as the regularization parameter tends to zero. We study the convergence of this scheme, both theoretically and numerically.
For more information, e-mail ma-chair@wpi.edu.
Boundary Layers of Incompressible Fluids with Navier Boundary Conditions
Friday, 2/15/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Yaguang Wang (Shanghai Jiaotong University & Northwestern University)
ABSTRACT:In this talk, we are interested in the small viscosity limit for the incompressible Navier-Stokes equations with the Navier friction boundary condition. First, by multi-scale analysis we formally study the asymptotic behavior of solutions when the viscosity goes to zero for several different cases of the slip length depending on the viscosity, and derive problems of boundary layers, secondly, we rigorously study the stability of boundary layers for the incompressible Navier-Stokes equations with anisotropic viscosities by using energy method, when the slip length is larger than the square root of the vertical viscosity, in which even though the boundary layer appears in the lower order of solutions but it yields the vorticity of fluids being unbounded in the viscosity limit.
For more information, e-mail ma-chair@wpi.edu.
Mathematics of Molecular and Cellular Biology
Friday, 2/29/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Roger Lui (WPI)
ABSTRACT: The study of molecular and cellular biology includes topics from DNA and RNA all the way to cell movement and chemotaxis. Currently, this is a very active branch of mathematical biology and many models are proposed in scientific journals every month. By and large, people don't know how to solve these problems because they are extremely difficult or because people don't know how to model them. In this talk, I shall illustrate these two points by discussing three examples taken from protein folding, biochemical network (cell signaling), and cell motility. I shall keep the technicality to a minimum.
For more information, e-mail ma-chair@wpi.edu.
March 2008
Portfolios and Risk Premia for the Long Run
Friday, 3/7/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Paolo Guasoni (Boston University)
ABSTRACT: This paper develops a method to derive optimal portfolios and risk-premia explicitly in a general diffusion model, for an investor with power utility and in the limit of a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk.
In models with several assets and a single state variable, long-run portfolios and risk-premia admit explicit formulas up the solution of an ordinary differential equation, which characterizes the principal eigenvector and its corresponding eigenvalue of a elliptic operator. Multiple state variables lead to a partial differential equation, which is solvable for most models of interest.
For each value of the relative risk aversion parameter, the paper derives the long-run portfolio, its implied risk-premia and pricing measure, and their performance on a finite horizon.
For more information, e-mail ma-chair@wpi.edu.
Levi L. Conant Lecture Series presents Jeffrey Weeks
Monday, 3/24/2008 3:00 PM-4:00 PM
Olin Hall, 107
SPEAKER: Jeffrey Weeks, Recipient of the 2007 AMS Levi L. Conant Prize
TITLE: The Shape of Space
ABSTRACT: When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a "multiconnected universe."
Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our universe.
The only prerequisites for this talk are curiosity and imagination. Middle school and high school students, people interested in astronomy, and all members of the WPI community are welcome to attend.
Coffee and tea will be available one half hour before the lecture.
For more information, e-mail ma-chair@wpi.edu.
Gelification and wax deposition in crude oils
Friday, 3/28/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Antonio Fasano (Departments of Mathematics U. Dini, University of Florence, Italy)
ABSTRACT: Waxy crude oils are mineral oils with a large content of heavy hydrocarbons(making what we call "wax"), normally dissolved. Various phenomena may occur when they are cooled down to sufficiently low temperatures: solid wax is segregated in the form of suspended crystals, the crystals may entangle trapping the oil and making a gel, the existence of thermal gradients may induce the migration of the still dissolved wax. This complex of phenomena may result in the formation of a solid deposit at the cold wall. The typical situation is the one of a submarine pipeline and the final consequence can be the pipe obstruction by the deposit.
A mathematical model will be presented describing the cooling process of a
static oil, combining all the previously mentioned effects. The model consists of a system of partial differential equations of various nature and contains several unknown interfaces. Existence of a solution is proved and qualitative properties are discussed.
For more information, e-mail ma-chair@wpi.edu.
Cardiac Systems Biology and Multi-Scale Modeling (Jointly with BE department)
Monday, 3/31/2008 12:00 PM-1:00 PM
Higgins Laboratories, 218
SPEAKER: Andrew McCulloch (BE Dept, UCSD)
ABSTRACT: Computational models of the heart can be integrative in several important ways. First, they permit information integration of genome scale data sets that would otherwise be difficult to interpret and understand. We illustrate this in the analysis of high-throughput phenotypic and metabonomic data on cardiac function in the fruitfly drosophila melanogaster. We use drosophila as a model organism for studying cardiac hypoxia tolerance, and integrating these data with genome-scale metabolic models. Systems models of cardiac myocytes achieve functional integration, by predicting how the functions of individual network components combine to give rise to physiological functions. We illustrate this with our recent models of -adrenergic regulation of myocyte excitation-contraction coupling. Multi-scale computational models aim to achieve structural integration across physical scales of biomedical organization from molecule to organize. We illustrate this with examples from our recent research on arrhythmia mechanisms in genetic disease and the effects of external interventions such as pacing and pericardiectomy on ventricular-vascular coupling in vivo.
For more information, e-mail ma-chair@wpi.edu.
April 2008
Coping with Uncertain Volatility when Pricing Equity Options
Friday, 4/4/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gunter Meyer (Georgia Institute of Technology)
ABSTRACT: In finance the volatility of an asset value is a measure of its randomness. It has the same influence on pricing options as the diffusivity has on the temperature in heat conduction. Volatility is not observable and we shall outline some attempts at estimating or bounding it from market data. Then we shall discuss the derivation and numerical solution of the nonlinear Black Scholes Barenblatt equation, a Partial-Integro-Differential equation, which yields upper and lower bounds on options prices for assets following a jump diffusion process with uncertain volatility. The basic tools are the maximum principle for the heat equation and a two point value problem solver based on the Riccati transformation.
For more information, e-mail ma-chair@wpi.edu.
Periodic nonlinear Schroedinger equation and photonic crystals
Friday, 4/11/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alexander Pankov (College of William and Mary)
ABSTRACT: We discuss the problem of existence of finite energy solutions to stationary periodic nonlinear Schroedinger equations. As an application we prove the existence of gap solitons in sertain models of one- and two-dimensional photonic crystals. (A photonic crystal is an artificial optical medium that has spatially periodic structure).
For more information, e-mail ma-chair@wpi.edu.
Math Awareness Week Lecture
Thursday, 4/17/2008 3:00 PM-4:00 PM
Higgins Laboratories, 218
SPEAKER: Tommy Ratliff, Wheaton College
TITLE: A Little Geometry, a Little Linear Algebra, and a Lot of Insight into Voting
ABSTRACT: The 2000 Presidential election taught many people that the winner of the popular vote may differ from the winner in the Electoral College. However, this is only a small example of the inconsistencies that are possible: If there are more than two candidates, then the voting procedure can have as much impact on the outcome as the preferences of the voters. We usually think of voting in the context of political elections, but the same framework used to understand elections can give insights into any group decision process.
We will see that a little geometry and linear algebra can go a long way toward explaining many voting paradoxes and how to avoid them.
Pizza & soda will be provided.
For more information, e-mail ma-chair@wpi.edu.
Creating materials with a desired refraction coefficient
Friday, 4/25/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Alexander G. Ramm (Kansas State University)
ABSTRACT: Many-body scattering problem is solved asymptotically when the size of the particles tends to zero and the number of the particles tends to infinity.
A method is given for calculation of the number of small particles and their boundary impedances such that embedding of these particles in a bounded domain, filled with known material, results in creating a new material with a desired refraction coefficient.
The new material may be created so that it has negative refraction, that is, the group velocity in this material is directed opposite to the phase velocity.
Another possible application consists of creating the new material with some desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle. In this application it is assumed that the incident plane wave has a fixed frequency and a fixed incident direction.
An inverse scattering problem with scattering data given at a fixed wave number and at a fixed incident direction is formulated and solved.
For more information, e-mail ma-chair@wpi.edu.
May 2008
Adaptive Finite Element Discretization of a Phase Field Model of Brittle Fracture
Friday, 5/9/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Christoph Ortner (Oxford University)
ABSTRACT: Adaptive finite element methods are a popular tool for numerical analysts for the efficient discretization of PDEs with singular or "near singular" solutions. In recent years, significant progress has been made in their analysis, and we now have a good understanding of their convergence and optimality.
In this talk, I will begin by describing adaptive finite element methods and their analysis at a simple model problem and then show how these ideas can be used to efficiently discretize the Ambrosio--Tortorelli functional, a "phase-field model" of brittle fracture. If time permits, I will present a convergence proof for an adaptive optimization algorithm for this highly non-convex functional.
For more information, e-mail ma-chair@wpi.edu.
Yaxiang Yuan--CANCELED
Friday, 5/16/2008 11:00 AM-12:00 PM
Stratton Hall, 203
For more information, e-mail ma-chair@wpi.edu.
Quasistatic crack growth in elasto-plastic materials
Tuesday, 5/20/2008 11:00 AM-12:00 PM
Stratton Hall, 203
SPEAKER: Gianni Dal Maso (SISSA, Trieste, Italy)
ABSTRACT: We present a variational model of quasistatic crack growth in elasto-plastic materials, which fits the general framework of the energy formulation of rate independent evolution problems developed by Mielke. The dissipation distance takes into account the effects of plastic flow and crack production. The model predicts that the crack tips belong to the closure of the plastic region.
For more information, e-mail ma-chair@wpi.edu.