For inquiries concerning the seminar send email to Jacek Szmigielski, szmigiel@math.usask.ca.

The seminar takes place in McLean Hall rm 242.1 on Wednesdays at 3:30 till 5:00 unless advertised differently. The main themes in the academic year 2007/2008 are: Pade Approximants and Magnetotelluric Inverse Problem.

**Next meetings**:

** Previous meetings**:
September 27, 2006 ,
October 4, 2006 .
October 11, 2006 .
October 18, 2006 .
October 25, 2006 .
November 1, 2006 .
November 8, 2006 .
November 15, 2006 .
November 22, 2006 .
November 29, 2006 .
January 3, 2007 .
January 10, 2007 ,
January 17, 2007 ,
January 24, 2007 ,
February 1, 2007 ,
February 8, 2007 ,
February 22, 2007 ,
March 1, 2007 ,
March 29, 2007 ,
April 19, 2007 ,
April 20, 2007 ,
October 1, 2007 ,
October 10, 2007 ,
October 17, 2007 ,
October 24, 2007 ,
October 31, 2007 ,
November 7, 2007 ,
November 13, 2007 ,
November 14, 2007 ,
November 21, 2007 ,
November 28, 2007 ,
January 9, 2008 ,
January 16, 2008
January 23, 2008
February 27, 2008
May 13, 2008

** Desiderata: **
It is envisioned that the seminar will serve two purposes:

(1) exposition of research topics of current interest

(2) training of participating undergraduate and graduate students

It is proposed that the seminar will consist of thematic blocks run by participating faculty. So far the following blocks have been identified:

** James Brooke**: Canonical realizations of symmetry groups in classical
mechanics - an application to classical Galileian (non-relativistic)
and Lorentzian (relativistic) mechanics ref CLASSICAL DYNAMICS: A
MODERN PERSPECTIVE E C G Sudarshan, N Mukunda John Wiley (1974) ISBN
0-471-83540-4 QA845.S8

** George Patrick**: Modern classical mechanics and the FPU
(Fermi-Pasta-Ulam computer experiment) based on the thesis of B. Rink:
Geometry and dynamics in Hamiltonian latices with applications to the
Fermi-Pasta-Ulam problem.

** Jacek Szmigielski**: Creation of shocks in fluid dynamics; the notion of
weak solutions to PDEs, and the survey by P. Lax (Abel prize recipient
in 2005) The formation and decay of shock waves, Amer.Math. Monthly,
79; 227-241, 1972.
If time allows and there is a sufficient interest, part of the second
semester could be on random matrix theory.

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Nonlinear partial differential equations; ODEs perspective.

The speaker will review the use of ordinary differential equations in solving partial differential equations. After a brief review of the separation of variables method for linear partial differential equations we will discuss a class of nonlinear partial equations (integrable PDEs) which finds numerous applications in areas as diverse as particle physics, optical fiber communication, nonlinear water wave theory, or pure mathematics to name just a few. This class has been intensively studied for more than 40 years, starting with the work of M. Kruskal and N. Zabusky on the so called Korteweg-deVries equation (KdV) describing among other things shallow water waves, followed by seminal works of C. Gardner, J. Greene, M. Kruskal, R. Miura (Plasma Physics Laboratory at Princeton University) and P. Lax (Courant Institute of Mathematical Sciences) establishing a profound connection between KdV and the scattering theory of 1 dimensional Schroedinger equation (thus an ODE problem). One of the more recent interesting additions to that class is the Camassa-Holm equation (CH) which has non-smooth particle-like solutions, called peakons. This equation appears to be linked to two types of ODE landscapes. As an example of an integrable PDE, CH is connected to a scattering problem akin to the one used for the KdV. As if this were not enough CH has distinct features reminiscent of yet another interesting class of nonlinear PDEs having an ODE connection -the quasilinear equations. The second half of the talk will deal with a review of the method of characteristics (ODEs) used to solve quasilinear equations. The main objective of the talk will be to elucidate the conceptual underpinnings of the PDE-ODE interplay.

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Nonlinear partial differential equations; ODEs perspective.

One of the more recent interesting additions to the class of integrable PDEs is the Camassa-Holm equation (CH) which has non-smooth particle-like solutions, called peakons. This equation appears to be linked to two types of ODE landscapes. As an example of an integrable PDE, CH is connected to a scattering problem akin to the one used for the KdV. As if this were not enough CH has distinct features reminiscent of yet another interesting class of nonlinear PDEs having an ODE connection -the quasilinear equations. This talk will deal with a review of the method of characteristics (ODEs) used to solve quasilinear equations. The review will be done on a conceptual level, no formal proofs will be given, however a few examples will be discussed. The geometric aspects of the method will be emphasized. The speaker will also discuss the rudiments of the concept of weak solutions. This latter topic is a preparation for a discussion of a paper by P. Lax on shock solutions.

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Theory and numerics of stability transitions for falling spinning underwater vehicles

Because of the presence of noncompact symmetry, the stability of the motion of a bottom heavy falling, spinning underwater vehicle, in the Kirchhoff approximation, is extremely delicate and complicated. If the vehicle falls too quickly, then this motion is spectrally unstable. As the speed is reduced, there is a (spin dependent) transition to spectral stability, and full nonlinear stability can be demonstrated by Kolmogorov-Arnold-Moser confinement. As the speed is further reduced, there is a (spin independent) transition to Energy-momentum stability. In the gap between the transitions, it is expected on general grounds that the addition of arbitrarily small dissipation will induce spectral instability. Since the EM transition is spin independent, and the stability analysis is general for Euclidean symmetry, the implication for gyroscopically stabilized devices is startling: in the presence of noncompact symmetry, robust stability may not be achievable by the use of spin. The stability analysis depends on the language, results, and insights, of modern mechanics set in the context of symplectic geometry. The numerical validation of the results ought therefore to preserve this context. In the seminar, the Kirchhoff approximation will be reviewed, the stability analysis will be summarized, and the numerical method used to simulate the system will be presented.

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Discrete Cubic String and Peakons

The theory of the Degasperis-Processi equation is very closely related to the spectral theory of a generalized string problem of order three (cubic string) . The speaker will describe two types of string problems which play a role in the analysis of special, weak solutions called peakons. These two problems generalize the Dirichlet and Neumann string to the cubic case. The spectral and inverse spectral problems in cases relevant to peakons have been recently solved by generalizing Stieltjes' technique of analytic continued fractions to dimension two. The solution involves in an interesting way a subset of matrices with nonnegative entries (totally nonnegative matrices). A few basic theorems about totally nonnegative matrices will be reviewed. No prior knowledge of these topics is required. This is a part of joint work with Hans Lundmark (Linkoeping University, Sweden) and with a UofS former undergraduate student (Engineering Physics) and speaker's 2003, 2004 NSERC summer student (currently at the University of Toronto pursuing her MSc degree), Jennifer Kohlenberg.

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The formation and decay of shock waves

A brief review of solving first order partial differential equations will be followed by the development of shock solutions to a single conservation equation. The PDE studied is quasi-linear and assumed to be genuinely non-linear, and the concept of weak solutions naturally arises due to the integral relation on which the PDE is based. The presentation is an exposition of a survey paper by P. D. Lax.

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The formation and decay of shock waves

Continuing Tyler's talk on the formation of shock solutions for a single conservation equation, I will discuss the decay of shock solutions. Specifically, the asymptotic behaviour of shock solutions will be examined, as well as the behaviour of periodic initial data (time permitting). This will cover most of the second half of the survey paper by P.D. Lax.

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Does a Brownian particle equilibriate?

The conventional equations of Brownian motion can be derived from the first principles to order lambda^2=m/M, where m and M are the masses of a bath molecule and a Brownian particle respectively. I discuss the extension to order lambda^4. The motivation for doing that is related to our interest in nolinear dissipation appearing beyond the lowest approximation. For the momentum distribution we have got an equation whose stationary solution is inconsistent with equilibrium Boltzmann-Gibbs statistics. This property originates entirely from nonlinear non-Markovian corrections which are negligible in lowest order but contribute to order lambda^4.

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**
David Callele
**

Math for Millions

The typical student, or member of the public at large, often believes that mathematics is the domain of a bunch of unusual people locked away in some corner of a university some place. And, for the realms of (relatively) pure mathematics (whatever that might be), they might be right. However, math doesn't just occur at universities. Applying mathematical techniques to solve tangible problems is a very active practice within the business world. Individuals with expertise in fields such as probability, statistics, computational geometry, and machine learning (to name but a few) are very hot commodities right now - as long as they can adapt to the industrial environment. In this talk, I am going to describe what it is like to do "Math for Millions", math when millions of dollars of investment capital are on the line, math when multi-million dollar decisions are made based on your equations. I will focus on the techniques used in "production mathematics" to try to give the audience a greater appreciation for the experience of doing math when the answer *must* be right. I will give high-level examples of the types of problems I am asked to solve and identify some of the skills that I need to achieve my goals within this environment.

David is an electronics engineer and computer scientist with 20 years of professional experience. Familiar with almost every aspect of the operations of customer-focused organizations committed to the delivery of superior high-technology products and services, David specializes in helping the customer determine what they really need, then delivering it. David's diverse experience includes telecommunications (core and access), packet-switching, computer-telephony integration, video game development, software engineering, and new product R&D. Currently a member of the team at Solido Design Automation, David has been an employee, an employer, a consultant, an entrepreneur, and an academic.

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A Problem-Solving Environment for the Solution of Nonlinear Equations

Nonlinear equations occur in many areas of science and engineering. The process of solving these nonlinear equations is generally difficult, from verifying the existence and uniqueness of solutions to finding a good initial guess that leads to a desired solution. In practice, Newton's method is the only mature and efficient method for solving a system of nonlinear equations. Many variants of Newton's method exist. However, the process of forming and choosing a suitable variant of Newton's method is complex. > > Many high-quality software libraries are available for the numerical solution of nonlinear equations. However, these libraries do not allow the user to conveniently form and experiment with different variants of Newton's method. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis proposal describes a problem-solving environment for studying the effects (e.g., performance) of different strategies for solving nonlinear equations numerically. It provides the researcher, teacher, or student, with a flexible environment for rapid prototyping and experimentation. Users can directly influence the solution process on several levels, such as specifying different termination criteria for the Newton iteration, experimenting with different methods for computing the Newton direction, specifying different storage types of the Jacobian matrix, as well as experimenting with different globalization strategies to converge to a desired solution. Thus, users are able to search for a suitable variant of Newton's method within the environment and detect numerical difficulties in the solution process as early as possible. Users may then easily transfer the prototype code to a high-performance environment.

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ON DYNAMICAL MODELING OF STOCHASTIC PROCESSES

Stochastic differential equations (SDEs) involving noisy terms are the natural tool to describe dynamics of a system interacting with the surroundings. Statistical properties of the noise are usually postulated in a more or less ad hoc way. However in many cases of practical interest the phenomenological approach to construct a SDE is inapplicable, as I discussed in the previous talk. Therefore it is important to understand how a proper SDE can be derived directly from the underlying dynamics, expressing the noise as an explicit (but complicated) function of dynamical variables of the surrounding thermal bath. I shall discuss two approaches to the problem based on the projection operator technique.

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STOCHASTIC DYNAMICS: PROJECTION OPERATORS, CONTINUOUS FRACTIONS, AND ALL THAT

I will continue to discuss application of the projection operator method to describe stochastic dynamics in complex systems. The method allows to derive an exact closed equation for a targeted variable which has a form of non-Markovian Langevin equation. This equation involves a correlation function of a "random" force which a priori is not known. One can apply the procedure again to derive a Langevin equation for the random force itself. This equation involves even more complicated correlation function. Proceeding in this way one can express relevant quantities in the form of continuous fraction, which will be a focus of the talk.

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Electromagnetic instability in a moving plasma as an eigen-value problem.

Plasma immersed in the equilibrium magnetic field supports a special type of oscillations (eigen-modes) (involving both plasma motion as the fluid and perturbations of the electromagnetic field) called Alfven waves (after Hannes AlfvNin (1908-1995) , 1970 Nobel Prize in Physics, http://public.lanl.gov/alp/plasma/people/alfven.html). When plasma is moving in the equilibrium the mode frequency is shifted by the Doppler shift. It is shown that such a system leads to an interesting type of the eigen-mode problem which has the eigen-frequencies complex, hence corresponding to the perturbations exponentially growing in time. Physically, it means that that plasma motion across the magnetic field leads to the generation of the Alfven wave perturbations. It is shown that such system is unstable with respect to the Alfven wave excitation. The relevant eigen-mode problem will be formulated. The variational, some exact and WKB solutions will be presented.

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Funding Opportunities in Applied Mathematics through MITACS

The Mathematics of Information Technology & Complex Systems (MITACS) welcomes you to attend a brief information session on the funding opportunities available to Professors or Graduate Students pursuing applied research in the Mathematical Sciences. The topics covered will include: The Network Centres of Excellence Program and MITACS A Brief Overview of MITACS Initiatives The MITACS Internship Program: Benefits for Universities, Professors & Graduate Students Procedural/eligibility and general FAQ How to work with MITACS to find industrial partners Relevant Case Studies Question & Answer period.

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Trajectory tracking of flexible link manipulators

Rigid link manipulators are heavy (load-carrying capacity only 5% of their weight) and consume considerable power. On the other hand flexible link manipulators (FLM) have smaller size and mass, lower peak power requirement and energy use, and their application in industry are expected to increase in the future, provided their performance becomes more predictable and reliable (follow a given path with no vibration). The derivation of a dynamic model for a FLM which captures the effects of the link$(B!G(Bs flexibility is an important step in model based research. Because a FLM is a continuous and deformable system, it has infinite degree of freedom (DOF). Usually a descritization method is used for dynamics; this descritization transfers the system from infinite DOF to a finite one, which is called truncated dynamic model. Moreover, due to the link$(B!G(Bs flexibility, the established control strategies studied for the rigid link manipulator is not applicable to FLM. The non-minimum phase characteristic of the dynamic equations and the existence of the non-holonomic constraints are among the main barriers in the controller design procedure. In this seminar first the dynamic model derivation of the FLM will be explained. A simple method to derive the closed form truncated dynamic equations of a FLM based on the truncated dynamic equations of a single flexible link manipulator will be discussed. Also a new approach for inverse dynamic calculation of a linear single flexible link manipulator will be presented. Finally the design of a new end-effector controller based on the singular perturbation technique will be discussed and the result of the simulation and experimental studies will be presented.

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Title: Observations on greedy composite Newton methods

The only robust general-purpose numerical methods for approximating the solution to systems of nonlinear algebraic equations (NAEs) are based on Newton's method. Many variants of Newton's method exist in order to take advantage of problem structure; it is often computationally infeasible to solve a given problem without taking some advantage of this structure. Moreover, it is generally impossible to know a priori which variant of Newton's method will be optimal for a given problem. In this talk, I will define the concept of a composite Newton method, i.e., a sequential combination of Newton variants, for solving NAEs. I will also describe a strategy for automatically generating a composite Newton method. The strategy is based on a greedy principle that updates the current iterate at regular intervals according to the best performing Newton variant from a number of variants running in parallel. It turns out to be possible for composite Newton methods to outperform optimal classical implementations of Newton's method, i.e., ones that only use one Newton variant on a given problem.

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Title: Novel Approach to Operator Splitting as Applied to Navier-Stokes Equations

Numerical methods for solving the Navier-Stokes equations have been developed very intensively in recent decades. Despite that, the optimal choice of specific numerical strategies for the discretization of the equations in time and space is still being sought. The vectorial operator splitting reduces, by orders of magnitude, the number of operations per iteration in comparison with application of direct solvers. In contrast with other splitting schemes, the vectorial operator splitting is applicable to the multidimensional case. In particular, the developed new schemes allow investigation of solutions of the Navier-Stokes equations at high Reynolds numbers. To meet the requirements of speed and accuracy, sophisticated numerical techniques must be implemented and efficient algorithms must be used in the simulations of such problems. Potential impact of results from the proposed research includes a wide class of applications in many disciplines.

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** David K. Goldak
info@empulse.ca **

Title: Transient Audio-Magnetotelluric surveying: Data Processing, Case Histories

Thunderstorm activity produces large amounts of electromagnetic energy which is trapped within the earth-ionosphere waveguide. This naturally occurring energy essentially consists of two parts; a low-level quasi continuous component and a transient component. The much larger transient component is due to relatively nearby, or equivalently, very large individual lightning discharges while the continuing component is due to the random sum of near global activity. Due to its$(B!G(B larger amplitude, the best possible signal-to-noise ratio is obtained through a time localized recording of the transient component. However, the transient events are strongly linearly polarized, the polarization diversity of which can affect the estimation of earth response curves. This was the motivation behind the development of our Adaptive Polarization Stacking (APS) algorithm, namely, a method which properly senses the polarization diversity of the data, the signal-to-noise ratio and sample size to return solid parameter and error estimates. The efficacy of our approach is shown in the results of surveys over a resistive buried valley extending to < 100 m depth and a graphitic conductor at > 700 m depth. Further verification in these cases is provided through good agreement with EM-47, UTEM III and gravity data respectively.

B.Sc. Engineering Physics, 1994, M.Sc. (Physics), 1998, University of Saskatchewan. While writing his M.Sc. thesis, the author worked at Barringer GeoSystems and subsequently Lamontagne Geophysics. After completing his M.Sc. under Dr. Ken Paulson, the author launched EMpulse Geophysics and began the design and construction of SFERIC-EM hardware and associated software, with the help of Peter Kosteniuk and Dr. Shawn Goldak. At present, we continue to improve our instrumentation and software and are actively engaged in many research projects.

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** Charles Cuell
(Ph.D. Candidate, Dept. of Mathematics and Statistics) **

Title: Aspects of Lagrange--d'Alembert Integrators

Lagrange--d'Alembert integrators are numerical simulations of Lagrangian systems that are obtained by a direct discretization of fundamental physical principles. This is in contrast to standard numerical integration methods that discretize the equations of motion. Lagrange--d'Alembert integrators inherit the natural physical properties of the continuous system whereas standard methods typically do not. For example, a Lagrange--d'Alembert simulation of a conservative system exhibits long time bounded energy error whereas a standard method will typically show a linear in time increase in the error. In this talk I will introduce Lagrange--d'Alembert integrators, give some examples to illustrate the theory and compare some results with standard methods.

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** G.W. Patrick
(Dept. of Mathematics and Statistics) **

Title: Variational structure recognition for nonholonomic mechanics

Classical holonomic mechanical systems, and discrete analogues of those, and also classical field theories, admit well known variational formulations. Solutions of these systems are critical points of an action functional subject to a fixed-boundary condition. But classical holonomic mechanical systems also have a well known symplectic formulation. This can be derived from the corresponding variational principle using a simple procedure, which only refers to generic concepts, such as "solution", and "boundary". The same procedure can be applied in a variety of contexts as a way or recognizing or identifying analogues of symplectic structures. This idea is useful and compelling in discrete analogues of these systems, and can be used to derive symplectic integration algorithms. So what happens when the procedure is applied, for the purpose of recognizing analogues of the symplectic structure, to nonholonomic systems, which are known not to be, in general, symplectic? Reference: Variational development of the semi-symplectic geometry of nonholonomic mechanics. Accepted Rep. Math. Phys., 42pp. http://math.usask.ca/~patrick/PatrickGW-2005-3.pdf

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** Ned Nedialkov
(Dept. of Computing and Software, McMaster University, Hamilton, ON) **

Title: McMaster University, Hamilton, ON

An interval method for initial value problems (IVPs) in ordinary differential equations (ODEs) has two important advantages over approximate ODE methods: when it computes a solution to an ODE problem, it (1) proves that a unique solution exists and (2) produces rigorous bounds that are guaranteed to contain it. Such bounds can be used to ensure that this solution satisfies a condition in a safety-critical calculation, to help prove a theoretical result, or simply to verify the accuracy and reliability of the results produced by an approximate ODE solver. We overview interval methods and software for IVP ODEs, discuss applications of these methods, and present the author's interval ODE solver VNODE-LP. Computing rigorous bounds in IVPs for differential-algebraic equations (DAEs) is a substantially more challenging task than in ODEs. A promising approach is Pryce's structural analysis combined with Taylor series methods. We outline this approach and discuss recent developments.

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** Tyler Helmuth
(Dept. of Mathematics and Statistics) **

Title: A Bit About Magnetotelluric Inversion

In this talk I will outline a little bit about magnetotelluric (MT) inversion, which is a method of geophysical surveying that is based upon measurements of naturally occurring electromagnetic fields. There is a close relationship between the MT inverse problem and another problem, called the inverse string (IS) problem. The IS problem will be explored, and along the way we'll run into Pade approximants and continued fractions. Some of the numerical challenges involved in the solution of the IS problem will be highlighted. The talk will be based upon work I've performed for a MITACS internship with EMPulse Geophysics.

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** Ray Spiteri
(Dept. of Computer Science ) **

Title:Stiffness Detection in Initial-Value ODEs

Stiffness is one of the most enigmatic yet pervasive concepts in the numerical solution of initial-value ODEs. This is especially true for ODEs arising from method-of-lines discretizations of PDEs. The goal of stiffness detection methods is to allow software to automatically choose a stiff or non-stiff integrator, where appropriate. Making the proper choice of integrator often has a profound effect on the overall efficiency of the software. However, the many faces of stiffness have resulted in many definitions, many disagreements, and many methods for its detection. In this talk, I will compare three methods for stiffness detection and show their performance on a representative sample of test problems.

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** Megan Lewis,
(Dept. of Mathematics and Statistics(undergraduate student in Mathematical Physics)) **

Title: Continued Fractions in Number Theory

A paper entitled Analytic Number Theory and Approximation, written by Walter Van Assche, will be gone through in detail by several undergraduate students. This paper covers rational approximation of irrational numbers, irrationality proofs and transcendence proofs. The rational approximation is typically done using continued fractions or Pad-Ai approximation. The paper also discusses the applications of rational approximation and special functions in analytic number theory. The first talk will cover the construction of simple continued fractions, as well as prove several useful properties of continued fractions.

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** Ryan Dean (joint work R. Spiteri)
(Dept. of Computer Science) **

Title: On the performance of an implicit-explicit Runge-Kutta method in models of cardiac electrical activity

Mathematical models of electric activity in cardiac tissue are becoming increasingly powerful tools in the study of cardiac arrhythmias. Considered here are mathematical models based on ordinary differential equations (ODEs) that describe the ionic currents at the myocardial cell level. Generating an efficient numerical solution of these ODEs is a challenging task, and in fact the physiological accuracy of tissue-scale models is often limited by the efficiency of the numerical solution process. In this talk, I will discuss the efficiency of the numerical solution of 4 cardiac electrophysiological models using implicit-explicit Runge-Kutta (IMEX-RK) splitting methods. I will discuss numerical experiments showing that variable step-size implementations of IMEX-RK methods (ARK3 and ARK5) that take advantage of Jacobian structure clearly outperform the methods commonly used in practice.

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** Megan Lewis and Fred Sage
(Dept. of Mathematics and Statistics (undergraduate students) **

Title: Continued Fractions in Number Theory

A paper entitled /Analytic Number Theory and Approximation/, written by Walter Van Assche, will be gone through in detail by several undergraduate students. This paper covers rational approximation of irrational numbers, irrationality proofs and transcendence proofs. The rational approximation is typically done using continued fractions or Padi approximation. The paper also discusses the applications of rational approximation and special functions in analytic number theory. The second talk will consist of several examples reinforcing the material covered in the previous talk on van Assche's paper, followed by the statement and proof of a theorem relating quadratic irrationals to periodic continued fractions. A link between the Fibonacci sequence and continued fractions will be discussed briefly.

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** R. Phillip Bording, Ph.D.
Husky Energy Chair in Oil and Gas Research
Memorial University of Newfoundland
St. John$-1rys, Newfoundland
**

Title: Wave Equations and Imaging of the Elastic Earth

The recent rise in global energy prices has made the average person wonder what the future holds for oil and gas supplies. In this talk I will illustrate the methods used to image the Earth and how complex the subsurface geological realm truly is. The basic mathematics for solving the Earth imaging problem has been known for some time, but the reality of the computing task far exceeds that of todayrys computers. We solve the problem today by taking short cuts and making assumptions. But can we continue to do so in the future is a serious question? The financial risk of mega-projects in the energy industry requires more exacting answers to just what is in the subsurface, and the time honored tradition of drilling to get the answer could be and is too expensive for our future. We must find more powerful mathematical tools and computer methods to extract every bit of knowledge from our data. So we examine new methods and machines like the IBM Blue Gene parallel computer, the cell, and GPUrys. Using the latest technology is a tradition in the energy industry and we are now challenged again to make it work.

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**
Artur Sowa
(Dept. of Mathematics and Statistics) **

Title: Building simple models for analysis of complex quantum systems

As technology pushes the barrier of nanoscale, it is important that we develop the art of constructing adequate models of quantum phenomena encountered in nanosystems. It is hardly ever that a very accurate model of a complex physical system can be constructed. At the same time, one may be perfectly able to control such systems by using simple schematic models that only capture some of their essential characteristics. This calls for an approach to modeling that utilizes the first principles of physics, but possibly goes beyond them in search of good practical models. I will present a general novel approach to building relatively simple models of quantum phenomena. The point of departure for the discussion will be the quantum mechanical density matrix and the Heisenberg equation. However, we will promptly redirect our attention to some novel structures. This path will lead us to a class of nonlinear differential equations relevant to the description of two-dimensional electron gas, particularly those phenomena which involve phases of electrons. Prerequisites: linear algebra, ordinary differential equations, basic quantum mechanics.

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Title: Analytic Number Theory and Approximation

A paper entitled /Analytic Number Theory and Approximation/, written by Walter Van Assche, will be gone through in detail by several undergraduate students. This paper covers rational approximation of irrational numbers, irrationality proofs and transcendence proofs. The rational approximation is typically done using continued fractions or Pad-Ai approximation. The paper also discusses the applications of rational approximation and special functions in analytic number theory. The initial portion of the talk will consist of the completion of the proof from last time (periodic continued fractions and quadratic irrationals) The second portion of the talk will focus on irrationality proofs. If time permits, both the proof of the irrationality of e (the base of the natural logarithm) and the proof of the irrationality of pi will be given.

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Title: Pade Approximants

A paper entitled /Analytic Number Theory and Approximation/, written by Walter Van Assche, will be gone through in detail by several undergraduate students. This paper covers rational approximation of irrational numbers, irrationality proofs and transcendence proofs. The rational approximation is typically done using continued fractions or Padi approximation. The paper also discusses the applications of rational approximation and special functions in analytic number theory. Basic properties of Pade approximations will be explained.

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Title: The linking probability of two self-avoiding polygons in a tube

In this talk, we present a model of a pair of polymers by two self-avoiding polygons (2SAP) which span a tubular sub-lattice of the simple cubic lattice. We consider the linking number of the 2SAP as a measure of whether the two polygons are linked. We show that the linking probability of 2SAPs approaches one as the size of the 2SAP, n, goes to infinity. It is also established that the linking number of a 2SAP is at most linear in n.

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Title: Proof of the irrationality of ln(2)

This talk will continue the detailed study of the paper entitled Analytic Number Theory, by Walter Van Assche. The focus of the presentation is the proof of the irrationality of ln(2), using sequences, a basic proof concerning irrationality, as well as the Prime Number Theorem.

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Title: Hermite-Pade Approximation

Abstract: This talk will continue the detailed study of the paper entitled Analytic Number Theory, by Walter Van Assche. The main focus of the talk will be Hermite-Pade approximation, an important extension of Pade approximation discussed last semester. We will derive formulas for remainders and a variety of orthogonality conditions implied by Hermite-Pade approximations of specific systems of Markov functions.

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** Time: 3:30 pm **
** Place: Arts 146 **

**Speaker: Bruce Davidson, Hydrologist
Environment Canada**

Title: Environmental Prediction - Coupled Hydrologic-Land-Surface-Atmospheric Models

Abstract: The Hydrometeorology and Arctic Laboratory (HAL) of Environment Canada in Saskatoon is a research and development lab dedicated to the improvement of water cycle prediction models and the appropriate use of their simulations. We work in collaboration with environmental scientists, project managers and computer specialists to develop and test experimental versions of numerical weather prediction, land-surface and hydrologic models. We also work with a variety of hydrologists and others interested in our model output to improve access to the information derived from the models. The presentation will introduce the work done at HAL and emphasize some of the software techniques used to improve both collaboration and environmental prediction.

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** Time: 3:30 pm **
** Place: Thorv 105 **

**Speaker: Harrell Sellers, IBM
About The Speaker
Harrell Sellers obtained his Ph.D. in Theoretical Chemical Physics from the University of Arkansas in 1979. He has worked at the University of Texas at Austin, Louisiana Tech University, and South Dakota State University. He was a PACER research Fellow, Minnesota Supercomputing Institute, University of Minnesota, Minneapolis from 1986 to 1988 and Head of the Computational Chemistry and Biology Group at the National Center for Supercomputing Applications, Beckman Institute, Univ. of Illinois from 1988-1992. Harrell is currently an Applications Specialist at IBM, a position he has held since 2003.
**

Title: Modeling of Catalytic Reactors: Carbon Dioxide and Methane Chemistry and Carbon Sequestration on Palladium

In this presentation we will discuss the development of computational models of catalytic reactors and an important application that is presently ongoing. To develop a computational model of a large set of chemical reactions on a metal catalytic surface we need to 1) develop the set of possible reactions that will be occurring on the surface, 2) determine the Arrhenius rate constant parameters required for the set of differential equations that describe the rates of the chemical reactions, 3) numerically integrate the set of chemical reaction rate equations (over time), and 4) optimize the conditions under which the reactor will run. In this presentation we briefly discuss the methods for creating a computational model of the catalytic reactor and present comparisons between computed parameters and experimentally derived ones. Also discussed are the steps in creating the computational model The example that we discuss is the application of these techniques to the methane (CH4) and carbon dioxide (CO2) chemistry on Palladium. This rich chemistry involves reactions that are suitable for carbon sequestration, methane conversion (converting methane to methanol, CH3OH), synthesis gas production (synthesis gas is a mixture of CO and H2), H2 production, formaldehyde (H2CO) production, and others. A unique aspect of this chemistry is the use of carbon dioxide as the carbon and oxygen source and methane primarily as a hydrogen source. Understanding this chemistry is important in the global effort to sequester atmospheric carbon and to the development of re-use strategies for CO2.

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