




Graduate Student Poster Session / Présentations des
étudiants diplômés (Sue Ann Campbell, Organizer)
 LINDSAY ANDERSON
Pricing a fixedstrike 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 difficulttostore 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 closedform
solution; the FixedStrike 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 CrankNicolson 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 
C_{0} z

 
å
s

C_{s} 
æ è


1 zp_{s}

+ 
1 p_{s}


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 smallscale 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), 1327.
2 Jihad Touma and Scott Tremaine, A map for eccentric orbits in
triaxial potentials. University of Toronto preprint, 1998, 129.
(This paper is dedicated to the memory of Dr. Roberto Mendel (195595),
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 interiorexterior 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 KarushKuhnTucker 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 predictorcorrector method in the second algorithm. The
approximate solution obtained gives a dual and a pseudofeasible primal
points. Since the primal solution is non feasible, a new pseudogap
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 HamiltonJacobi (HJ) 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 HJ equation.
Using the formal link between HJ equations and conservation laws for
the solution gradient, we have designed a gradientpreserving second
order accurate, linearly stable, numerical scheme to compute the
solution to the cellproblem. 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 Relementary
derivations of the polynomial ring in three variables over R. A
consequence of the main result is that the kernel of every
k[X_{1},¼,X_{n}]elementary monomial derivation of
k[X_{1},¼,X_{n}][Y_{1},Y_{2},Y_{3}] 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 neuronsv1, 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 qth derivative of a function f is the
only proper way to study IVP of differential equations.
D^{q}f = 
ó õ

x
0


(ts)^{b} G(1b)


d^{n+1}f(s) ds^{n+1}

ds 

where q = n+b, 0 £ b < 1, n is an integer.
Consider the linear fractional differential equation
a_{1}D^{a1}x(t)+a_{2}D^{a2}x(t)+¼+a_{n}D^{an}x(t) = f(t) 

To define the initial value problems, let
a_{k} = m_{k}+b_{k}, 0 £ b_{k} < 1, m_{k} are integers 

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

a_{1}D^{a1}x(t)+a_{2}D^{a2}x(t)+¼+a_{n}D^{an}x(t) = 0, t > 0 
 x(0) = x_{0}, x¢(0) = x_{1},..., x^{(M)} = x_{M1} 

 

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 wellunderstood only if the
socalled 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.
 KITSUN 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 secondorder 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
Selfcycling fermentation and impulsive dynamical systems

Selfcycling fermentation is a computeraided process used in the
treatment of waste products in sewage plants and toxic waste cleanup.
Systems are noncontinuous, 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.

