2008 Fall Semester.

• 11/14/2008, Labate, Sparsity in mathematics and engineering.

In many problems from mathematics and engineering, a function or a signal is broken up as a superposition of elementary components. A classical example of such approach is the Fourier series representation. A representation is sparse if any function in a certain class can be well approximated using only a few terms in its representation. This property is particularly desirable in a number of applications such as data compression and denoising. In this seminar, I will describe more precisely what are the benefits of sparse representations, how to properly "measure" the sparsity of a representation, and go over some recent developments in this field.

 

• 11/07/2008, Zhigang Zhang ,An in silico protocell with contracting FtsZ rings.

I will introduce an in silico protocell, which is a mathematical model designed to understand the periodic behavior of living cells, especially the cell self-replication. It includes a chemical reaction network which simulates the intracellular reactions. Different from other minimal whole cell models, we incorporate geometric properties, the cell?s surface area and volume, in the model. An FtsZ ring model is also included in the protocell, which simulate the assembly and contraction of the FtsZ ring located at the middle of the cell. We then determine the shape of the cell by minimizing the bending energy of the membrane with given cell surface area, volume, and ring size. Without the FtsZ ring, the cell grows but does not show division. Method to calculate the shape of the cell will also be introduced in the talk.

 

• 10/30/2008, David Blecher ,Characterizations in mathematics.

I have been asked to begin by repeating some advice I gave in the grad student seminar last year about finding a great research problem, and a little about approaching ones research. After that I plan to talk about characterizations in mathematics. An object that has very many very different characterizations, is usually one which is very important, with a rich and amazingly fertile theory. The different characterizations lead to very varied approaches to studying the object. I also want to talk about good characterizations, and bad ones, giving examples from my field, and from a couple of recent papers.

 

2008 Spring Semester.

• 04/28/2008, Bernhard G Bodmann , Random Rounding Rules in Redundant Representations.

In the linear world, rounding can be a terrible thing! This talk compares several round-off strategies for basis and frame expansions in Hilbert spaces. When the family of vectors used in the expansion is overcomplete, then independent rounding is inferior to other strategies such as sigma-delta quantization. We will also see that randomized rounding strategies can lead to an improved (average) behavior compared to deterministic rules. The exposition will be illustrated with audio examples. This talk is intended for a general audience.

• 04/18/2008, David Blecher , Some great conjectures.

I wanted to talk a little about how to go about finding a great research problem, a little about a couple of the best mathematicians in my field and what makes their research great (and how they approach their research, so that maybe we can strive to emulate them). Of course, like most speakers, I'd also like to a say a little in general terms about my latest papers.

 

• 03/14/2008, Cleopatra Christoforou , A Traffic Flow Model.

In this talk, we will formulate a mathematical model for traffic problems that is in terms of a partial differential equation. Our goal is to solve this equation and understand traffic phenomena that we experience in our daily commute!

 

• 02/29/2008, Alexandre Caboussat , Ordinary Differential Equations with Discontinuities: Applications in Air Quality Modeling.

Many processes in atmospheric chemistry can be modeled with ordinary differential equations that exhibit discontinuity points. The tracking of such points requires appropriate "event location" techniques when these points are not known a priori. We will start by reviewing some examples of ODEs with discontinuities and the corresponding numerical techniques. We will then apply these numerical methods to a problem in air quality modeling that couples ordinary differential equations and optimization problems, namely the computation of the chemical composition of atmospheric particles. We will show how to take advantage of both the optimization and differential equations features of the model to describe tracking methods for the accurate detection of the discontinuities in the evolution and we will emphasize the difficulties encountered when the events are not explicitly known.

 

• 02/15/2008, Pete Casazaa , Applications of Hilbert space frames.

Hilbert space frames have traditionally been used in signal/image processing. Recently, there have arisen a variety of new applications to wireless communication, internet coding, physics, Biomedical Engineering, speech recognition technology, and more. We will look at some of the new applications of frame theory and how frame theory has begun to impact some of the most famous unsolved problems in ``pure'' mathematics.

 

2007 Fall Semester.

• 10/05/2007, G. Auchmuty , Variational Methods for PDEs. .

Many of the important partial differential equations of geometry and physics can be described as the solutions of variational principles. That is their solutions are the critical points, usually minimizers, of a functional defined on a subset of a Banach space. Usually this functional has had a physical interpretation as an "energy" and the theory usually only applied to certain classes of time independent equations.

Recently the methods of convex analysis have enabled us to develop variational methods for characterizing the solutions of much more general classes of equations. This talk will describe some simple examples of such variational principles, including principles for evolution equations and "non- potential" equations.

 

• 09/07/2007, Papadakis , Images and Signals in UH-Math .

A variety of imaging applications including image acquisition, inverse problems and signal encoding are among the research interests of the speaker and of several of his colleagues. In this talk an overview of some of these topics will be presented. The key players are Simon Alexander, Robert Azencott, Bernhard Bodmann, Manos Papadakis and Vern Paulse from UH-Math and Donald Kouri from Chemistry-Physics. this talk is not intented to present in depth any of these fields of study, but rather to be a `sampler' of what can be `cooked' with a good mix of pure and applied mathematics. The applications that will be presented are in biomedical and seismic imaging

 

Previous Gradudate Student Seminas

• 04/13/2007, Field ,Chaos and Structure in Dynamics.

The talk is about how randomness can arise in deterministic dynamics and how we can measure and quantify randomness. We indicate how in systems which are statistically indistinguishable from fair coin tossing we can nonetheless see structure.

• 03/30/2007, Andrew Torok , Models for chaos: from Markov chains to Young towers .

Some systems that look chaotic can be described by a relatively simple symbolic dynamics (trajectories in a finite graph, with specified transition probabilities). These are the "uniformly hyperbolic" systems, considered in the 1960's by Dmitri Anosov and Stephen Smale. However, many chaotic systems exhibit only "non-uniformly hyperbolic" behavior. A model to describe these was introduced in the late 1990's by Lai-Sang Young. We will sketch these models, and discuss consequences one can derive from them.

• 03/02/2007, Mark Tomforde , Three Problems Motivated by Graph C*-Algebras.

I was asked to give a talk to graduate students about problems related to my research. I am going to discuss three problems which are motivated by my own work: the first is C*-algebraic in nature, the second deals with a question in combinatorics, and the third is a problem in algebra. These are open questions, and any graduate student who is interested could begin working on one of these problems with me.

• 02/16/2007, Prof. Canic , Mathematics and Cardiology: Partners for the Future.

The speaker will talk about several mathematical projects that stem from an interdisciplinary endeavor between the researchers in the Mathematics Department at the University of Houston, the Texas Heart Institute, Baylor College of Medicine, the Mathematics Department at the University of Zagreb, and the Mathematics Department of the University of Lyon 1. The projects are related to mathematical modeling and analysis of non-surgical treatment of aortic abdominal aneurysm and coronary artery disease using endovascular prostheses called stents and stent-grafts.

Through a collaboration between mathematicians, cardiovascular specialists and engineers several novel mathematical models have been developed to study blood flow in compliant (viscoelastic) arteries treated with stents and stent-grafts. The mathematical tools used in the derivation of the effective, reduced equations utilize asymptotic analysis and homogenization methods for porous media flows.

Analysis of the existence and well-posedness of a unique solution to the resulting fluid-structure interaction problem uses energy estimates and fixed point theorem techniques applied to a system of partial differential equations of mixed, hyperbolic-parabolic type defined on a domain with a moving boundary. A numerical method, based on the finite element approach, was developed, and numerical solutions were compared with the experimental measurements. Experimental measurements based on ultrasound and Doppler methods were performed at the Cardiovascular Research Laboratory located at the Texas Heart Institute. Excellent agreement between the experiment and the numerical solution was obtained.

Several research projects and open mathematical problems related to fluid-structure interaction in blood flow will be discussed.

This year marks a giant step forward in the development of the partnership between mathematics and medicine: the FDA (the United States Food and Drug Administration) has, for the first time, imposed a requirement that mathematical modeling and numerical simulations to be used in the development of peripheral vascular devices before an FDA approval can be granted. This research has been funded, in part, by the National Science Foundation, the National Institutes of Health, the Texas Higher Education Board (ARP) and by the St.Luke's Episcopal Hospital in Houston.

Collaborators on the projects include: Profs. R. Glowinski, T.W. Pan, G. Guidoboni, A. Mikelic, J. Tambaca, S. Lapin; Medical Doctors: D. Rosenstrauch, Z. Krajcer and C. Hartley, and Graduate Students: Taebeom Kim, Mate Kosor, and Jian Hao.

• 11/17/2006, Mark Tomforde , "The Academic Job Search ."

I'm going to talk about applying for academic jobs in the U.S. I'll give a timeline of all the stages in the job search, discuss the process of sending out applications and interviewing, and talk about what kinds of things one needs to do in the last > year of graduate school in order to get an academic job. I encourage people to come with questions.

This will be the last graduate seminar for the fall semester. The talk will be extremely useful to those of us who are looking for an academic career.

• 11/03/2006, Prof. Robert Azencott , "Random Gibbs Fields: Application to Texture Identification in Numerical Images. ."

Gibbs fields are stochastic models originally introduced by physicists to modelize the dynamics of large systems of small interactive magnets localized at the nodes of a 2D or 3D rectangular network of physical sites . 2D or 3D numerical images attach a positive “gray level” J(p) to each node “p” of a rectangular grid of pixels (or voxels) ”Textures” are images where the gray level function J is far too irregular to be modelized by smooth functions,and is viewed as superpositions of vibrating patterns at various scales Textures can efficiently be modelized by random Gibbs fields, for automated texture identification, as well as for computer graphics simulations.

• 09/08/2006, Prof. Glowinski , "Nonlinear Eigenvalue Problems."

The main goal of this lecture is to discuss the solution of a classical nonlinear eigenvalue problem from physics and show how it relates to some conjectures from the theory of nonlinear partial differential equations. The results of numerical experiments will complete this presentation.

• 08/03/2006, Matt Dulock, Basic Concepts in Riemannian Geometry.

Abstract: I will present the main tools used in differential geometry. I`will discuss manifolds, vector bundles, differential forms, connections, and`Riemannian metrics. At the end I hope to present the statement of some`interesting results including the Gauss-Bonnet theorem.

• 07/13/2006, Sonia Sharma, Real Operator Algebras.

Abstract : I will talk about operator algebras over the field of reals and we will to see the real version of the BRS (Blecher-Ruan-Sinclair) theorem, via the complexification of real operator algebras.

• 07/27/2006, Shikha Baid , "Application Of Wavelets to Medical Imaging (Part II)"

Abstract I will introduce the problem of analysing CT scans to identify various types of tissues. We first use a wavelet transform to seperate features and textures of different tissues. The encoded image is then analysed for textural consistensies with a localised version of statistical hypothesis testing. My emphasis will be on the wavelet transform.

• 06/29/2006, Mrinal Raghupathi , Function Theory and Groups.

Abstract: I will describe the notion of a covering map and the group of deck transformations and indicate how they are used to relate function theory on multiply connected regions in the complex plane to function theory on the disk. I will describe work done in the sixties by Frank Forelli and an application to the corona theorem.

• 06/13/2006, Charu Sharma, "Exact and Asymptotic Tests for the Analysis of Case-Control Studies "

Abstract: I will give a brief introduction to the chi-square test, Fisher's exact test. Cochran Armitage test which are used in categorical analysis. I will then discuss the application of these tests in the analysis of Case-Control studies of genetic markers.

• 04/28/2006, Saurabh Jain, "Imaging the Earth's Interior with Isotropic Wavelet Transforms "

Abstract: The oil exploration industry depends on efficient solution of the acoustic wave equation to image the interior of the earth. I will discuss the imaging technique call ‘Wave Equation Migration’ and our algorithm for the same based on the First Generation Isotropic Multiresolution Analysis (IMRA)

I will begin by explaining the concept of a Mutiresolution analysis and the associated wavelet frames. IMRA is and example of a Frame Multiresolution Analysis constructed by Manos Papadakis and Bernhard Bodmann in R^n which is non-separable in the sense that it is not a tensor product of one-dimensional wavelets. The elements of the frame arising from the IMRA are isotropic (translates of radial functions) and hence the images obtained using these do not have directional artifacts present in tensor product constructions

• 03/31/2006, Prof. Hasan Sayit, "On the different proofs of the Dalang-Morton-Willinger theorem."

Abstract: Roughly speaking, the Dalang-Morton-Willinger theorem states that for a finite sequence of price process, the no-arbitrage condition implies the existence of an equivalent martingale measure. We investigate different approaches to the proof of this theorem.

• 03/03/2006, Mary Flagg, "The Role of the Jacobson Radical of the Endomorphism Ring in the Baer-Kaplansky Theorem"

Abstract: The celebrated theorem by Baer and Kaplansky states that two torsion groups are isomorphic if their endomorphism rings are isomorphic rings. Analogous theorems have been proven for torsion-free and certain classes of mixed modules over a complete discrete valuation domain. The main tool employed in these isomorphism theorems is the use of the idempotent projections to identifiable direct summands.

The question under study is the extent to which the idempotents are necessary for the isomorphism theorems. The Jacobson radical of the endomorphism ring is an ideal of the ring that contains none of the nonzero idempotents. Yet Hausen, Praeger and Schultz proved that for torsion groups which are unbounded modulo their maximal divisible subgroup, two groups are isomorphic if only the Jacobson radicals of their endomorphism rings are isomorphic rings.

We extend this result to modules over complete discrete valuation domains. Isomorphism theorems will be presented for torsion free modules, torsion modules with an unbounded basic submodule and for certain classes of mixed modules of torsion-free rank one.

• 02/24/2006, Deepti Kalra, "Complex Equiangular Cyclic Frames"

Abstract: We derive various interesting properties of complex equiangular cyclic frames for pairs (n,k) using Gauss sums and number theory. We further use these results to study the random and burst errors of some special cases of complex equiangular cyclic (n,k) frames.

• 02/17/2006, Dr. Jiwen He , Convex envelope, primal-dual interior-point algorithm, and modeling atmospheric aerosol state
• 02/03/2006, Dr. V. Paulsen, Title: Quantizers, Sigma-Delta Algorithms and Frame Paths

Abstract:  Whenever real numbers, i.e., analog information, is converted to digital, a quantizer must be used. Formally, a quantizer is a map, Q:[-L,+L} --> [-L,+L] with finite range. The most often used quantizer is the 16-bit quantizer, which divides the interval into 2^{16} subintervals of equal size and assigns the midpoint of each subinterval as the value Q(x) and represents it by 16 bits. This makes the error |x-Q(x)|< 2^{-17}.

A "coarse" quantizer might only use 4 bits and have an error 2^{-5}, but has the advantage that one only needs 1/4 of the storage.

Every quantizer has the problem that even if each individual error is small, the sum of all the errors grows with the number of real values "sampled". Sigma-delta algorithms help correct this problem.

Recently, it has been realized that a similar problem happens when the data being quantized is really a vector in a high dimensional space. The small errors in each coordinate can lead to a large error in the norm of the difference between the original vector and the vector of quantized values.

This has lead to the idea of sigma-delta algorithms built around frames. Roughly, a frame is an overdetermined basis, so that computing all the frame coefficients of a vector leads to a new way to "oversample".

• 11/18/2005 Dr. M. Papadakis, Title: Inside arteries, and under the sea-bottom in the Gulf of Mexico: Common challenges.
  
  Abstract: One of the challenges in today's multidimensional data processing is the elimination of directional bias. Directional bias is inherent in many processes such as data compression and coding, identification of certain patterns and compromises the ability of algorithms to perform well in orientations other that vertical, horizontal and diagonal. On the other hand, nature's paradigms, retinal filtering and wave propagation reveal that isotropy is the rule rather than a processing by rows and columns. To address this problem we are working on non-separable multiresolution analysis based on frames. Our applications involve cardio-vascular imaging and modeling of wave propagation. In the first application, we are interested to detect vulnerable plaque in coronary arteries, plaque that raptures with no prior warning and kills; in the latter we are interested in the block-diagonalization of the phase-shift operator arising form the solution of the one-way acoustic wave equation and is used for years to model wave propagation for oil exploration.

• 11/3/2005, Bruce Lowe , Texas A & M University, An introduction to interest rate modeling.

  Abstract: A survey of the key mathematical ideas associated with the modeling of the evolution of the term structure of interest rates is provided. Particular attention shall be paid to the important contribution of Vasicek.

• 10/21/2005. Y. Wang , Patterns of synchrony in Lattice dynamical systems.

  Abstract:Dynamics of networks receive explosive attention recently. However, general studies of network dynamics are fairly rare. Stewart, Golubitsky, Pivato and Torok develop a general theory on dynamics of networks --theory on coupled cell systems. Lattice dynamical systems are coupled cell systems, which have been used in many application. In this talk, we consider patterns of flow invariant subspaces of lattice dynamical systems by using the general theory.

• 04/29/2005. Chakradhar Iyyunni , Fun with Balloons and Polymeric Cylinders : An analogue of Euler Buckling of Columns for pressurized cylinders.

  Abstract: Surely, everybody's blown into balloons and wondered about the evolution of its shape. Most balloons are membranes, i.e., one dimension is much smaller than the other two. I will explain how to analyze the behavior of a thick polymeric cylinder under internal pressure in the context of nonlinear elastostatics.

  I will begin with explanations of the relevant physics and principles from nonlinear elasticity theory. I will use the "buckling of columns" scenario as a caricature and illustrate the procedure for arriving at the eigenvalue problem of interest. Closed-form solutions are rare in nonlinear elasticity but we demonstrate one for a wide class of materials. If time permits, some motivation for the studying the relevant bifurcation problem (relying on Prof. Golubitsky's work) will be given.

• 04/22/2005. Dr. Bao, Time efficient paths in R2.

  Abstract: In this talk, we will discuss the notion of geodesics. Instead of distance minimising paths, we will adopt the perspective that geodesics are the most time efficient paths of travel. We shall see that this perspective gives rise to a rich variety of geometries, even on simple spaces such as R2. Common sense is the only pre-requisite for understanding this talk.

• 04/15/2005, Dr. Bernhard G Bodmann, Tomorrow's High-End Digital Audio: Sigma-Delta for Everybody

  Abstract: This talk explains a remarkably simple error compensation algorithm that is at the core of the new Super-Audio CD standard. We start with a brief review of the sampling theorem, a cornerstone of digital audio. Then we explore the problem of round-off errors, and how those errors are suppressed with the help of oversampling and the sigma-delta algorithm. Many illustrations and examples are included throughout the discussion.

• 04/8/2005, Dr. Morgan, Recreational Mathematics

  Abstract: Can you use your knowledge of basic mathematics to explain how an object rolls along a curve? We discuss some motion problems and pose some recreational problems...just for the fun of it!

• 04/01/2005, Dr. Torok, Understanding Chaos: The Lorenz attractor

  Abstract: Studying a simple ODE, Lorenz discovered in 1963 an object that is called today a strange attractor: nearby points are attracted to a set of fractal dimension, and move around this set chaotically.

  Understanding this attractor was one of the 18 problems for the twenty-first century proposed by Field medalist Steven Smale.

  Namely: Is the dynamics of the ordinary differential equations of Lorenz that of the geometric Lorenz attractor of Williams, Guckenheimer, and Yorke? Tucker answered this question in the affirmative in 2002. His technical proof makes use of a combination of normal form theory and validated interval arithmetic.

  The goal of this talk is to explain what it means to understand systems that look chaotic. Link to Dr. Torok's website for more information.

• 03/11/2005, Mr. Dekang Xu, UH, Proper holomorphic mappings.

  Abstract: In this talk, we will introduce the proper holomorphic mappings between balls in complex spaces with dimesion larger than 1. We will also give a new result about rational proper mappings with degree 2 from B^2 to B^4.

• 03/04/2005, D. Blecher, UH, An illustration from analysis of the principle of generalization

  Abstract: This talk will be partly some advice and comments about doing research in mathematics, and partly an example of a research project, which illustrates how mathematics builds and evolves. I will try to be quite nontechnical.

• 02/18/2005, Caboussat , UH, Numerical Simulation of Free Surface Flows: An Overview and Some Applications in Mold Filling

  Abstract:

  Free surface flows appear nowadays in many engineering or mathematical problems. Industrial processes such as casting, injection filling, blood flows or ink filling involve complex free surface phenomena that have to be solved using adapted approaches.

  We present first an overview of the methods and models that can be used for the modeling and numerical simulation of free surface flows. Then we present a numerical model for the simulation of complex three-dimensional free surface flows, in which an incompressible liquid interacts with a compressible gas in a cavity.

  A volume-of-fluid method is used to track the liquid domain. The method combines a time splitting scheme to decouple the physical phenomena and a two-grids method consisting in a regular structured grid of small cells and a finite element unstructured mesh of tetrahedra.

  We address some theoretical questions that can arise for free boundary problems and present numerical results in the frame of mold filling.

• 01/28/2005, Selwyn Hollis , Department of Mathematics, Armstrong Atlantic State University, Savannah, GA, Modeling Southern Pine Beetle Infestations,

  Abstract:

  The southern pine beetle (SPB) is a highly destructive species of bark beetle that inhabits pine forests in the southern United States and parts of Central America. In natural pine forests, the ecological role of the SPB presumably is to hasten the demise of weak or damaged trees, creating kindling for fires that contribute positively to the long-term health of the forest ecosystem. In commercial forests and developed residential areas, SPBs are responsible for hundreds of millions of dollars in timber losses annually. SPB populations exhibit outbreak behavior similar to other forest insect pests, typically persisting at low endemic levels for a number of years before rapid increase to epidemic levels and subsequent collapse. In this talk, we will describe (1) a simple ecosystem-level (ODE) model that successfully mimics outbreak behavior and (2) a new spatio-temporal model that potentially may be used to simulate the spatial spread of infestations.
• 10/22/2004, S. Canic , Modeling and mathematical analysis of blood flow in human arteries,

  Abstract:

  As you are reading this abstract your arteries are pulsating in response to the pressure waves caused by the contractions and relaxations of the heart muscle. In fact, your abdominal aorta, whose average radius is around 1.2 cm, might be experiencing a change in the radius of up to 10 percent during each cardiac cycle. Mathematically, this interaction between blood flow and ``compliant'' vessel wall can be modeled by a free-boundary problem describing the coupling between the incompressible Navier-Stokes equations modeling blood flow and a set of ``structure'' equations describing the behavior of vessel walls. Even when the simplest structure models are considered such as, for example, the Navier equations for a linearly elastic membrane, this fluid-structure interaction problem is so complex, that not only is the existence theory out of reach at the moment, but the numerical simulations of the 3D problem are exceedingly difficult. I will describe, in simple terms, the basic models used to study blood flow in human arteries, and the main mathematical difficulties behind their analysis. Movies showing numerical simulations associated with the treatment of certain cardiovascular diseases will be presented. Experiments performed at the Texas Heart Institute will be discussed.

  Collaborators on this research include cardiologists Dr. Krajcer and Dr. Rosenstrauch at the Texas Heart Institute, Prof. Josip Tambaca at the University of Zagreb, Croatia. Prof. Andro Mikelic at the University of Lyon 1, France, Prof. Ravi-Chandar at the University of Texas in Austin, Dr. Craig Hartley at Baylor College of Medicine, and students Joy Chavez, Serguei Lapin, Cynthia Chmielewski (UH) and Daniele Lamponi (EPFL).

• 10/15/2004, S. Ji From the Hilbert's seventeenth problem.

  Abstract:

  The talk will start from introducing the Hilbert's seventeenth problem. Then we should turn to another formulation in terms of complex analysis and present the Quillen-Cartlin-D'Angelo theorem. It will involve Real Analysis (real polynomial, Fourier series), Linear Algebra (positive definite matrix), functional analysis (linear transformations between Hilbert spaces, compact operator), and complex analysis (Bergman kernel). Some hard analysis part will be skipped. It will illustrate interaction among different branches of mathematics.

• 10/8/2004, R. Hoppe Numerical solution of optimization problems with PDE constraints.

  Abstract:

  The optimization of the shape and the topology of continuum mechanical structures by means of a systematic, physically consistent design methodology is referred to as structural optimization. The design criteria are chosen according to a goal oriented operational behavior of the structures under consideration. They typically lead to nonlinear objective functionals depending on the state variables describing the operational mode and the design variables determining the shape and the topology. The state variables satisfy systems of partial differential equations reflecting the underlying physical laws whereas technological aspects may give rise to further equality and inequality constraints on both the state and the design variables.

  After discretization the resulting nonlinear programming problem is usually solved by an alternating iteration where the discretized state equations and the optimization are carried out sequentially. Here, we advocate the use of recently developed ''all-at-once'' approaches. The characteristic feature is the numerical solution of the state equations as an integral part of the optimization routine. In particular, we focus on primal-dual Newton methods combined with interior-point techniques for an appropriate handling of the inequality constraints. Special emphasis is given on the efficient solution of the primal-dual Hessian system by taking advantage of its special block structure.

  Applications include

  • optimization of microcellular biomorphic ceramics,
  • shape optimization of electrorheological shock absorbers,
  • topology optimization of high power electromotors.

• 09/17/2004, K. Josic Dynamical Systems, geometric singular perturbation theory and neuroscience

  Abstract:

  The Hodgkin-Huxley equations model the propagation of an action potential down a nerve fiber, and represent one of the cornerstones of modern neuronscience. Due to the nonlinearity of the equations, it is necessary to look at their reductions to be able to study them analytically. I will discuss several reductions that have been particularly useful, and explain how they can be used in models of Parkinson's disease, memory, and other situations.

• 09/17/2004, V. Paulsen Frames, Graphs and Codes

  Abstract:

  In recent years the idea of building "analog", instead of just binary codes has risen in the signal processing community. The idea of an analog code is that given a k-tuple of real numbers that one wishes to transmit over a noisy channel, one should first encode it as an n-tuple of real numbers, n>k, transmit the n-tuple and then decode.

  The theory of frames is one way to geometrically describe such encoding-decoding situations. In our work, we have shown that when they exist the "optimal frames" are necessarily "equiangular". These equiangular frames can be constructed using special kinds of graphs and various "performance criteria" of the frames reduce to concrete questions of a graph theoretical nature.

  We will introduce this theory and discuss some of the still open problems in graph theory that arise from studying frames.

• 09/10/2004, Ilya Timofeyev Statistical Description and Predictability in Dynamical Systems.

  Abstract:

  Without going into too many technical details we will explain several commonly used mathematical terms, such as "statistical description", "turbulence", and "predictability". Common approaches to these issues in modern applied mathematics will be discussed and illustrated with some simple examples.

  As a part of this talk I will also discuss possible research projects in these areas with applications to fluid dynamics and atmospheric research. These projects are the result of collaboration of three faculty members

• 04/30/2004, Mr. Philip D Jacobs Symmetric Expanding Attractors

  Abstract:

  Attractors are topological structures which are useful in the study of dynamical systems. This talk will first sketch the development of expanding attractors from 1) the classical solenoid attractor of S. Smale to 2) generalized solenoid attractors by R. Williams to 3) the introduction of symmetry into Williams' structures by M. Field, I. Melbourne, and M. Nicol. The symmetry is by way of a finite group acting on 3-space. In the case of Field, Melbourne and Nicol, the group acts freely on the expanding attractor. I will then discuss recent developments to construct expanding attractors on which the finite group action is not free.

• 04/23/2004, Dr. Kao Alternative Approaches to Pricing Financial Derivative

  Abstract:

  In this talk, we give a brief survey of alternative approaches for modeling the pricing of contingent claims. The paradigms we consider include those of risk-neutral pricing models, dynamic equlibrium models, and pricing in incomplete markets.

• 04/02/2004, Mr. Pepper Binding Independence from Above

  Abstract:

  Starting with a short introduction to graph theory, an upper bound on the independence ratio of benzenoid systems will be presented. Implications of this result in mathematical chemistry will be discussed, including the independence-stability hypothesis for fullerenes and benzenoids. Some upper bounds for the independence number of arbitrary graphs will be given, including the annihilation number and the delta-residue, if time allows.

• 04/16/2004, Dr. Nicol Brownian motion and applications.

  Abstract:

  We briefly describe the mathematical theory of Brownian motion and give applications to PDES, chaotic dynamical systems and mathematical finance.

• 04/02/2004, Mr. Pepper Binding Independence from Above

  Abstract:

  Starting with a short introduction to graph theory, an upper bound on the independence ratio of benzenoid systems will be presented. Implications of this result in mathematical chemistry will be discussed, including the independence-stability hypothesis for fullerenes and benzenoids. Some upper bounds for the independence number of arbitrary graphs will be given, including the annihilation number and the delta-residue, if time allows.

• 03/26/2004, Prof. Ru The Amazing ABC Conjecture

  Abstract:

  In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer. Fermat's last theorem, for instance, involves an equation of the form x^n + y^n = z^n. More than 300 years ago, Fermat conjectured that the equation has no solution if x, y, and z are all positive integers and n is a whole number greater than 2. Andrew J. Wiles finally proved Fermat's conjecture in 1994. The Wiles effort could help point the way to a general theory of three-variable Diophantine equations. The key element appears to be a problem termed the ABC conjecture, which was formulated in the mid-1980s by Joseph Oesterle and David W. Masser. The conjecture offers a new way of expressing Diophantine problems, in effect translating an infinite number of Diophantine equations (including the equation of Fermat's last theorem) into a single mathematical statement. The statement of the ABC conjecture is amazingly simple compared to the deep questions in number theory, however, it turns out be equivalent to all the main problems. It's at the center of everything that's been going on. The ABC conjecture is considered to be the most important unsolved problem in Diophantine analysis In the past fifteen years, it was discovered that the theory of Diophantine analysis and the Nevanlinna theory in complex analysis bear surprising similarities. The counterpart of the ABC conjecture in Nevanlinna theory, called the Truncated Second Main Theorem, plays an crucial role. In this talk, I'll explain the ABC conjecutre and its consequences, as well as briefly describe its analogy in Nevanlinna theory.

• 03/12/2004, Prof. Pan Can we use computers to perform experiments?

  Abstract:

  In this talk we will discuss how well we can get from the computer simulation of the interaction between the particles and fluid. The models and methodologies will be discussed briefly.

• 03/05/2004, Prof. Torok Knots, Braids, and Operator Algebras

  Abstract:

  We will describe how seemingly unrelated fields, knot theory (low dimensional topology) and von Neumann algebras (a topic of Functional Analysis), were connected when Vaughan Jones discovered a new knot invariant. [In part for this work, Jones received the Fields Medal in 1990. A link about V. Jones is at here .]

• 02/20/2004, Prof. Keyfitz, Multidimensional Conservation Laws: How to Solve Hyperbolic Problems with Elliptic Methods

  Abstract:

  We begin with the basic classification of partial differential equations (PDE), in which many time-dependent problems are of hyperbolic type; their solutions are characterized by wave propagation, finite domain of dependence, and focussing of singularities. The mathematical tools (which will NOT be the focus of this talk) developed to analyse linear and semilinear hyperbolic equations do not help much with nonlinear problems. By contrast, in elliptic PDE (which typically govern time-independent problems), the quasilinear theory is a relatively simple variant of the linear theory. Despite impressive recent advances in the theory of hyperbolic conservation laws in a single space variable, there is little theory for multidimensional conservation laws. One approach to this problem is to study self-similar solutions. It turns out that one ends up with quasilinear elliptic equations --- and can take advantage of the advanced state of elliptic theory. The talk will conclude with some recent results of Suncica Canic, Eun Heui Kim and myself related to weak shock reflection and the von Neumann paradox.

• 10/17/2003, Mr. Serguei Lapin, EFFECT OF INTERSTITIAL GAS ON POWDER FLOW.

  Abstract:

  The problem of flow and consolidation of fine-grained bulk solids through a vessel comprised of cylinders, cones and a cylinder-cone combination is modeled in this paper. Unlike coarse-grained bulk solids, the pressure of the gases trapped between the solid particles is substantial when the powder is fine. Therefore, the flow of powder is treated as a two-phase interaction between gases and solids. To simplify the modeling of the problem, spatial averaging is adopted. This formulation, even though 1-D in space, includes geometry effects of the container. The model is derived using mass continuity and force equilibrium resulting in three partial differential equations that describe the density of the solid, the velocity of the solid and the pressure of the gas. Apart from the three partial differential equations, the model also contains an ordinary differential equation that describes the height of the powder in the vessel, or the free boundary, in time. Furthermore, a numerical method based on finite difference approximations in space and a Runge-Kutta method in time is implemented and computational results are presented to discuss the validity of this approach.

• 10/17/2003, Prof. S. Fajtlowicz , Stability and Expanding Properties of Fullerenes and Benzenoids.

  Abstract:

  Graffiti is a computer program whose early conjectures inspired results by many mathematicians including Alon, Bollobas, Chung, Erdos, Lovasz, and Kleitman. In the past few years the program made conjectures in carbon chemistry suggesting that stable fullerenes minimize their independence numbers and that they tend to be good expanders. A set of vertices of a graph is independent, if no two of them are adjacent. The independence number of a graph is the number of vertices of a maximum independent set. I will define the expanding properties of graphs and shortly discuss the origin of this concept (related to the subject of this talk) and its relation to some classical mathematical problems and problems in theoretical computer science. Neither the independence number nor the expanding properties were discussed before in carbon chemistry and initially some chemists were openly critical about Graffiti's stability conjectures. Nevertheless, thanks to extensive computation of Craig Larson, it appears now that there is a strong statistical evidence for the stability-independence hypothesis and that the stability-expanding hypothesis is consistent with the accepted theories of stability of benzenoids.

• 10/17/2003, Prof. Jutta Hausen, GENERALIZATIONS OF SOME CONCEPTS FROM LINEAR ALGEBRA.

  Abstract:

  Given vector spaces $V$ and $W$ over a field $F$, a linear transformation from $V$ to $W$ is a function $T: V \to W$ preserving vector addition and scalar multiplication in the sense that $T(\alpha +\beta) = T(\alpha)+ T(\beta)$ and $T(c\alpha)=cT(\alpha)$ for all $\alpha, \beta\in V$ and all scalars $c\in F$. The set of all linear transformations from $V$ to $V$ is a ring under the operations of pointwise addition and composition of functions which has the identity function on $V$ as multiplicative identity. These concepts generalize to the notions of (unital) $R$--module and $R$--module homomorphism where $R$ is a ring with a multiplicative identity element: the only difference is that the scalars need not come from a field, just from a ring $R$ with identity. Analogous to vector spaces, given an $R$--module $G$, the set of all $R$--module homomorphisms $\varepsilon: G \to G$ is a ring, $E_R(G)$, called the endomorphism ring of $G$. The talk will focus on settings in which the term $R$--module endomorphism coincides with with $R$--homogeneous function. A function $f: G\to G$ is said to be $R$--homogeneous if $f(ca)=cf(a)$ for all $a\in G$ and all $c\in R$. The set of all $R$--homogeneous functions from $G$ to $G$ is a near--ring, ${M}_R(G)$, containing the endomorphism ring of $G$. Usually, $E_R(G) \neq {M}_R(G)$. Several open problems will be presented.

• 09/23/2003, Prof. Herbert Amann, Control of heat conducting viscous flows.

  Abstract:

  In this talk I will explain how the motion of a viscous incompressible fluid - say, of water in a container heated through a net of wires - can be mathematically modelled. Then I will describe a strategy for rigorous mathematical proofs.

• 09/19/2003 (posponed), Prof. Keyfitz, Multidimensional Conservation Laws: How to Solve Hyperbolic Problems with Elliptic Methods

  Abstract:

  We begin with the basic classification of partial differential equations (PDE), in which many time-dependent problems are of hyperbolic type; their solutions are characterized by wave propagation, finite domain of dependence, and focussing of singularities. The mathematical tools (which will NOT be the focus of this talk) developed to analyse linear and semilinear hyperbolic equations do not help much with nonlinear problems. By contrast, in elliptic PDE (which typically govern time-independent problems), the quasilinear theory is a relatively simple variant of the linear theory. Despite impressive recent advances in the theory of hyperbolic conservation laws in a single space variable, there is little theory for multidimensional conservation laws. One approach to this problem is to study self-similar solutions. It turns out that one ends up with quasilinear elliptic equations --- and can take advantage of the advanced state of elliptic theory. The talk will conclude with some recent results of Suncica Canic, Eun Heui Kim and myself related to weak shock reflection and the von Neumann paradox.

• 09/12/2003, Prof. R. Glowinski, A FRICTION PROBLEM: MATHEMATICAL AND NUMERICAL ANALYSIS

  Abstract:

  The accurate simulation of remote manipulator systems like the articulated arms of the Space Shuttle and of the International Space Station requires taking dry friction into consideration. This can be done in a relatively simple way if one accepts to replace differential equations by differential inequalities. The resulting mathematical problems and their numerical analysis are sufficiently simple to be understood by senior undergraduate and 1st year graduate students with a good background in real and/or functional and/or applicable analysis. One of our motivations at presenting this dry friction problem is that it will give us an opportunity to "visualize" a weak convergence phenomenon, showing thus that this phenomenon can take place in real life related problems.

• 04/04/2003, Prof. Mike Field, The Structure of Deterministic Chaos

  Abstract:

  It is now well known that deterministic dynamical systems can behave "chaotically". But what does this actually mean and how can we measure it? In this talk we explore what is meant by terms such as "random" and show how simple deterministic systems can have statistics indistinguishable from coin-tossing. The talk will include a visual component where we will show some of the intricate structure that can be embedded within chaotic systems.

• 03/28/2003, Mr. Damon Hay, Noncommutative Topology and the Banach-Stone Theorem

  Abstract:

  The set of continuous complex-valued functions C(X) on a compact Hausdorff space X is an example of a commutative C*-algebra. We will examine how this algebra completey describes the topological structure of X, and vice versa. Major results in this direction include Gelfand's theorem and the Banach-Stone theorem. We will emphasize this connection between topology and algebra for its own beauty and use it to discuss some generalizations of the classical Banach-Stone theorem. This theorem demonstrates that the topological structure of X is encoded in the linear structure of C(X). Ultimately We hope to discuss some recent extensions of this result to the noncommutative setting of general C*-algebras.

• 03/21/2003, Prof. Manos Papadakis, Abridging Human Vision and Image

  Abstract:

  We will present some new challenging topics in wavelet design for Image Processing stemming form the study of human vision. We will focus on certain characteristics of human vision and show how they can be incorporated in the design of non-separable multi-scale analysis.

• 03/14/2003, Prof. Marty Golubitsky , Pattern Formation and Symmetry

  Abstract:

  There is a paradox concerning symmetry: symmetric causes can have asymmetric effects. This paradox is called spontaneous symmetry-breaking. In recent years researchers have shown that it plays a major role in pattern formation in physical systems. Chaos rides on the back of another paradox: deterministic mathematical models can produce random behavior. Finally, the way in which symmetry and chaos co-exist is a third paradox: symmetry suggests order and regularity while chaos suggests disorder and randomness. The combination leads to a striking series of pictures and to a notion of pattern on average.

• 02/28/2003, Prof. Jiwen He , Mathematical Modeling and Computation of Air Quality.

  Abstract:

  The mathematical models in air quality typically involve highly nonlinear coupled partial differential equations whose solution poses computational challenges. In this talk we review problems arising from important issues related to mathematical modeling of air quality, and computational methods to address them quantitatively. We present examples of scientific advances made possible by a close interaction between atmospheric science and mathematics, and draw conclusions whose validity should transcend the examples.

• 02/21/2003, Ms. Vrushali A Bokil , The Perfectly Matched Layer Technique for the Reflectionless Absorption of Electromagnetic Waves.

  Abstract:

  The numerical solution of partial differential equations in an unbounded domain requires the introduction of artificial boundaries to limit the region of computation. One needs boundary conditions at these artificial boundaries in order to guarantee a unique and well posed solution to the differential equation. In turn these boundary conditions are necessary to guarantee stable difference approximations. We would like these artificial boundaries and corresponding boundary conditions to affect the solution in a manner such that they closely approximate the free space solution that exists in the absence of these boundaries. In particular, one would like to minimize the amplitude of waves reflected from these artificial boundaries.

  The Perfectly Matched Layer (PML) is a technique of free space simulation for solving unbounded electromagnetic problems. This technique is based on the the use of an absorbing layer especially designed to absorb, without reflection, the electromagnetic waves. Designed by J. P. Berenger in 1994, the PML technique has been demonstrated to be the most effective free space simulation technique that has been developed for linear electromagnetic wave propagation so far.

  In this talk, I will describe the PML technique for two dimensional problems, and then discuss the numerical implementation of this model. Examples will be presented that demonstrate the effectiveness of the model.

• 02/14/2003, Prof. Giles Auchmuty , Optimization, Variational Principles and some basic questions in science.

  The talk is related to the materials covered in our courses on Applied Analysis, PDEs and Optimization theory.

  Abstract:

  Many basic questions in mathematics and science take the form "What is the optimal way to ...", or "What is the state of a system which minimizes some energy (or related) functional?" For example the famous 9 volume "Course on Theoretical Physics" by Landau and Lifshitz made an effort to describe most topics in terms of variational principles governing the behavior of the system under consideration.

  It turns out that variational principles not only provide good motivation for deriving physical laws and relationships but they lead to good mathematical formulations of the problems that lead to analytical results and good methods for numerical computation and simulation.

  In this talk I will illustrate this by describing the mathematical formulation of two basic scientific problems whose mathematical analysis is far from complete. These analyses are very important as they resolve issues that have arisen in the development of numerical methods and computational codes for simulation of these problems.

  The first problem is to characterize the equilibrium state of a chemical reaction involving gaseous reactants. This is an issue in problems of pollution, devising good fuels and in the petrochemical industry. The second problem is what data is needed to solve Maxwell's equations of electromagnetism in a region with various types of boundaries? Attempts at numerical solution of Maxwell's equations have made us realize how many basic mathematical questions about the equations have not been solved.