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Graduate Student Poster Session / Présentations des étudiants diplômés
(Sue Ann Campbell, Organizer)

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LINDSAY ANDERSON
Pricing a fixed-strike asian call option with continuous arithmetic averaging

Asian options are options for which payoffs are related to the average value of the underlying asset over a specified time period of the options's life. They are of great interest for hedging purposes to ``natural'' buyers or sellers of difficult-to-store commodities such as electrical power. Because of the path dependence of these options they are more difficult to value than the standard vanilla options. In fact, for some specific cases of Asian options, there is no closed form solution.

We consider the problem of pricing an option with no closed-form solution; the Fixed-Strike Asian call option with Continuous Arithmetic Averaging. The solution of this problem is given in terms of the inverse Laplace transform of a confluent hypergeometric function. We value the option using numerical inversion of this hypergeometric function as well as with a Crank-Nicolson Scheme and a Monte Carlo approach. We investigate the stability and rate of convergence of the three numerical methods.

WAEL BAHSOUN, Department of Mathematics and Statistics, Concordia University, Montreal, Quebec  H4B 1R6
Invariant measures for inner functions

This thesis describes the chaotic behavior of inner functions in the unit disk and in the upper half plane. Absolutely continuous invariant measures for inner functions, ergodicity and exactness will be discussed. Moreover, We will prove that for a class of meromorphic functions represented in the form

g( z) = A+e é ë Bz- C0 z - å s  Cs æ è 1 z-ps + 1 ps ö ø ù û

(1)

where A,B,C_s, p_s are real constants, B,C_s ³ 0 and e = ±1. Then, the Julia set J(g) = R if and only if the restriction of g to R is ergodic.

SHARENE BUNGAY, Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland  A1C 5S7
Optimization of first order saddle points using genetic algorithms

Optimization has applications in most scientific areas and many algorithms currently exist for the optimization of extrema on a surface. However, few techniques for the determination of first order saddle points have been developed.

In computational chemistry the minima of a potential energy surface represent the reactants, products, and intermediates of a reaction, while saddle points represent transition state structures. The fleeting existence of transition states makes it difficult to determine the structure of these compounds experimentally. Knowledge of these transition state structures are important in constructing reaction mechanisms, hence computational methods for their optimization are desired.

The computational technique of genetic algorithms has recently been applied to optimization problems and shows potential in the determination of saddle points. There, the genes of individuals are encoded with the parameters of the problem to be solved, and through successive biased breeding the evolving population converges to the best solution with the desired features of the optimum. Employing genetic algorithms for the optimization of first order saddle points, as opposed to extrema, poses fundamentally different problems which must be carefully addressed, not the least of which is the determination of a suitable fitness function for the preferential convergence to saddle points.

A discussion of the genetic algorithm technique, the problems it presents when applied to saddle point optimization, and how these problems are addressed, is presented.

ANDREA DOESCHAL, University of Western Ontario, London, Ontario  N6A 5B7
Assessment of a cellular model for sediment transport in braided rivers

Despite of the enormous progress of computational fluid dynamic models in recent years, little success has been achieved in understanding the essential dynamic components in braided rivers using sophisticated numerical flow models. As an alternative to the traditional modeling technique, an extremely simple model for water and sediment transport in a cohesionless river bed has managed to produce realistic braided patterns. This model uses ideas from cellular automata theory and relies only on information about the bed topography.

We have assessed the performance of this cellular model by applying it to data from a physical small-scale model of a braided river. Estimates for the input parameters were derived from established flow and sediment transport formulae or from computational experiments and distributed sensitivity analysis was carried out. For efficiency, the cell resolution of the computational grid was altered and the model's response to varying grid dimensions was studied. Various visual and quantitative methods were applied to assess the modeling results.

The results of the computational experiments confirm the necessity of an uphill and lateral flow and sediment transport component in braided rivers. Without these components, the river bed eventually reaches a steady state with only minor topographic changes. The modeling results also indicate an unrealistic water distribution as the main cause for poor predictions of the evolution of the river bed. For future work, modifications in the water transport rules are advocated.

JOHN JAN DROZD, Department of Applied of Mathematics, The University of Western Ontario, London, Ontario  N6A 3K7
The apsidal angle and its derivative in Newton's apsidal precession theorem

Newton's apsidal precession theorem in Proposition 45 of Book I of the Principia has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the many applications of this theorem. We have examined the apsidal angle q(N,e), where N is the index for the centripetal force law, for varying eccentricity e. The apsidal angle turns out to be dependent only on e and N. The logarithmic potential (N = 1) has turned out to be of particular interest in consequence of the recent interest in luminosity and density functions of galaxies and the Hubble Space Telescope photometric observations relevant for such functions. We study the apsidal angle for this potential. The resulting integral is an interesting improper integral with singularities at both limits. It is simpler to perform a numerical integration by computer after the integration is done analytically to the maximum possible extent. We use a ten point gaussian quadrature routine and Maple plots to determine the limiting roots of integration.

References

1 Sree Ram Valluri, Curtis Wilson, William Harper, Newton's Apsidal Precession Theorem and Eccentric Orbits. J. Hist. Astronom. 28(1997), 13-27.
2 Jihad Touma and Scott Tremaine, A map for eccentric orbits in triaxial potentials. University of Toronto preprint, 1998, 1-29.

(This paper is dedicated to the memory of Dr. Roberto Mendel (1955-95), Professor of Applied Mathematics at the University of Western Ontario, killed in an automobile accident at the height of his career.)

KHALID ELYASSINI, Department of Mathematics and Computer Science, University of Sherbrooke, Sherbrooke, Quebec  J1K 2R1
A new resolution approach for linear programming problems

We describe two interior-exterior algorithms for linear programming problem. The algorithms are based on path following idea and use a two parameter mixed penalty function. Each iteration updates the penalty parameters. An approximate solution, of Karush-Kuhn-Tucker system of equations which characterizes a solution of the mixed penalty function, is computed by using only one Newton direction in the first algorithm and by predictor-corrector method in the second algorithm. The approximate solution obtained gives a dual and a pseudo-feasible primal points. Since the primal solution is non feasible, a new pseudo-gap definition is introduced to characterize primal and dual solutions. Finally, Some numerical results will be presented.

BOULAEM KHOUIDER, Numerical combustion via an asymptotic flamelet library
Université de Montréal, Montréal, Quebec

A rigorous asymptotic theory for turbulent premixed flames has been recently proposed by Majda and Souganidis, using ideas from viscosity solutions and the homogenization of Hamilton-Jacobi (H-J) equations. We use this theory to model the subgrid effects in large eddy simulations (LES) of premixed flames. According to the homogenized problem, the effect of the small scales are described via a nonlinear eigenvalue problem, the cell problem, which is a first order H-J equation. Using the formal link between H-J equations and conservation laws for the solution gradient, we have designed a gradient-preserving second order accurate, linearly stable, numerical scheme to compute the solution to the cell-problem. The scheme is robust, efficient and accurate, and has been used to parameterize the effect of turbulence on the flame speed via an asymptotic flamelet library for a variety of small scale flows. Convergence and accuracy of this new method will be discussed, as well as its performance when applied to a representative idealized turbulent flame.

JOSEPH KHOURY, University of Ottawa, Ottawa, Ontario  K1N 6N5
On some properties of elementary derivations in dimension six

Given a UFD R containing the rationals, we study R-elementary derivations of the polynomial ring in three variables over R. A consequence of the main result is that the kernel of every k[X₁,¼,X_n]-elementary monomial derivation of k[X₁,¼,X_n][Y₁,Y₂,Y₃] is finitely generated over k by linear elements in the Y_i's (k is a field of characteristic zero). In particular, seven is the lowest dimension in which we can construct a counterexample to Hilbert's fourteeth's problem of Robert's type.

SEHJUNG KIM
Dynamical behaviors of a network of three neurons with a piecewise linear signal function

We consider three neurons-v1, v2 and v3 such that v1 and v2 excite each other, v2 and v3 inhibit each other, and v1 and v3 interact via v2. We use a piecewise linear signal function for theoretical study, and a sigmoid signal function for numerical simulations. We consider long term behaviors of this network.

We obtain at most three equilibria for this system and classify these equilibria as thpe 0, I, II and III. We find that types I, II and III are always stable and only type 0 can be unstable. When type o is stable, no other types of equilibria can appear. When type 0 is unstable, it appears with one of other types of equilibria. In addition type I, II, and III cannot exist together. Simulation for a sigmoid signal function further confirms the same observation.

XIAORANG LI, Department of Applied Mathematics, The University of Western Ontario, London, Ontario  N6A 5B7
Fractional differential equation with maple

There have been quite a few definitions for fractional derivatives. However, most of them are not suitable for studying initial value problems of differential equations. Davison and Essex found out that the following definition of q-th derivative of a function f is the only proper way to study IVP of differential equations.

Dqf = ó õ x 0  (t-s)-b G(1-b) dn+1f(s) dsn+1  ds

where q = n+b, 0 £ b < 1, n is an integer.

Consider the linear fractional differential equation

a1Da1x(t)+a2Da2x(t)+¼+anDanx(t) = f(t)

To define the initial value problems, let

ak = mk+bk,    0 £ bk < 1,    mk are integers

and let

M = max 1 £ k £ n  mk

then we can show there exists a unique solution for the IVP

a1Da1x(t)+a2Da2x(t)+¼+anDanx(t) = 0,    t > 0 x(0) = x0, x¢(0) = x1,..., x(M) = xM-1

If all the derivatives are of rational orders, and the right hand side f(t) is limited to polynomials, exponential functions and trignomeric functions, we can use the Laplace transform to find the analytical solution of the initial value problem. The solutions can be expressed by hyper geometric functions. A Maple program has been developed to solve fractional differential equations. This software provides a convenient tool for studying fractional calculus.

(Joint work with C. Essex)

RAMIN MOHAMMADALIKHANI, University of Toronto, Toronto, Ontario
Moduli spaces of flat connections on surfaces

I explain some alternative definitions of these muduli spaces and their equivalence. Then I give some physical motivation for studying these spaces through gauge theory. I also try to make clear the relation of the subject with symplectic geometry and other branches of geometry and topology that are used to answer questions on these spaces.

ISRAEL NCUBE, Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario  N2L 3G1
Stochastic approximation for a simple automatic classifier via a dynamical systems approach

Stochastic approximation deals with the problem of characterisation of the long term behaviour of discrete stochastic algorithms. For example, does the algorithm converge to a fixed point? One of the primary areas where this problem arises is in the study of unsupervised neural learning algorithms. This problem is well-understood only if the so-called associated ordinary differential equation (ODE) possesses exactly one globally asymptotically stable equilibrium point. In this case, it is known that, under some fairly reasonable assumptions, the stochastic algorithm converges, with probability one, to the equilibrium point of the associated ODE. However, if the ODE possesses multiple locally stable equilibria, nothing is currently known about the convergence of the algorithm to any specific one of these equilibria.

In this presentation, we outline an algorithm that attempts to address this problem.

KIT-SUN NG, University of Toronto, Toronto, Ontario  M6J 2E6
Quadratic spline collocation methods for systems of PDEs

We consider Quadratic Spline Collocation (QSC) methods for solving systems of two linear second-order PDEs in two dimensions. Optimal order approximation to the solution is obtained, in the sense that the convergence order of the QSC approximation is the same as the order of the quadratic spline interpolant.

We study the matrix properties of the linear system arising from the discretization of systems of two PDEs by QSC. We give sufficient conditions under which the QSC linear system is uniquely solvable and the optimal order of convergence for the QSC approximation is guaranteed. We develop fast direct solvers based on Fast Fourier Transforms (FFTs) and iterative methods using multigrid or FFT preconditioners for solving the above linear system.

Numerical results demonstrate that the QSC methods are fourth order locally on certain points and third order globally, and that the computational complexity of the linear solvers developed is asymptotically almost optimal. The QSC methods are compared to conventional second order discretization methods and are shown to produce smaller approximation errors in the same computation time, while they achieve the same accuracy in less time. This is joint work with Prof. Christina Christara.

LILA RASEKH, University of Guelph, Guelph, Ontario
Discrete cosine transform

I am working on numerical linear algebra.My talk is going to be about Discrete cosine Transform (DCT), which is widely applied in engineering.In my talk I am going to present a recursive algorithm for DCT with the structure that shows the generation of the next higher order DCT from two identical lower order DCT's. As a result, this method requires fewer multipliers and adder than other DCT's algorithm.

ROBERT SMITH, McMaster University
Self-cycling fermentation and impulsive dynamical systems

Self-cycling fermentation is a computer-aided process used in the treatment of waste products in sewage plants and toxic waste cleanup. Systems are non-continuous, but improve significantly on other similar methods such as batch fermentation and the chemostat.

The process can be modelled as a system of ordinary differential equations with impulsive effect. The theory of impulsive dynamical systems provides mathematical predictions for the fermentation process, allowing for improved applications in industry.