


Contributed Papers Session / Communications libres (Org: WRS Sutherland, Dalhousie University)
 CLAUDIA DANIELA CALIN, University of Alberta, Edmonton, Alberta T6G 2G1
Coagulation equation with unbounded kernels, particle sources
and sinks

During the past few years, increasing attention and effort have been
given to the mathematical theory of coagulation equations which model
the formation of large particles by the coalesence of smaller
particles. Coagulation equations arise in a number of problems in the
physical and polymer sciences, colloid chemistry, aerosol physics.
One interesting aspect of the coagulation equation, that occurs for
certain coagulation kernels, is that mass need not be conserved for
all time. The phenomenon whereby conservation of mass breaks down in
finite time is known as gelation and is physically interpreted as
being caused by the appearance of an infinite "gel" or
"superparticle".
 MEGAN DEWAR, Carleton University, Ottawa, ON K1S 1S2
Universal Cycles and Block Designs

The term Universal Cycle was introduced by Chung, Diaconis and Graham
in 1992. A Universal Cycle (Ucycle) is defined as follows: Let
F_{n} be a family of combinatorial objects of rank k
with m = F_{n}. Let A be a fixed alphabet from which
the elements of each F Î F_{n} are selected. A Ucycle
for F_{n} is a cycle (x_{0}, ..., x_{m1}), where
x_{i} Î A (for i=0,...,m1) and where every element of
F_{n} appears exactly once as some kblock of this
cycle. The most wellknown type of Ucycle is for the nbit binary
numbersthis is a De Bruijn cycle of order n. In this talk we
discuss what is known about the existence and construction of Ucycles
for various combinatorial objects. We begin with a look at the
published research which focusses on Ucycles for ksubsets of
nsets and kpermutations of nsets. We then look at Ucycles
for block designsa family of combinatorial objects previously not
considered.
 ZHIPENG DUAN, Memorial University of Newfoundland, P. O. Box 40, Faculty of
Engineering, St. John's, NL A1B 3X5
Impingement AirCooled Plate Fin Heat SinksPressure Drop Model

The performance of impingement aircooled plate fin heat sinks differs
significantly from that of parallel flow plate fin heat sinks. A
semiempirical zonal model has been published to predict impingement
flow in plate fin heat sinks. This model is not convenient to apply
since it involves a large number of equations. A simple impingement
flow pressure drop model based on developing laminar flow in
rectangular channels is proposed. To test the validity of the model,
experimental measurements of pressure drop are performed with heat
sinks of various impingement inlet widths, fin spacings, fin heights
and airflow velocities. It was found that the predictions agree with
experimental data within 20 numbers less than 1200. The simple model is suitable for parametric
design studies.
 JEANETTE EDGE, Memorial University of Newfoundland
Fully Developed Flow of NonNewtonian Fluids in Non Circular
Ducts

There has been much research conducted on the behavior of Newtonian
fluids in the process industry for pipes of various cross sectional
shapes. Shear stress related correlations have been derived for the
elliptical, rectangular, polygonal, and annulus shaped ducts. It has
been shown that for a given value of aspect ratio the dimensionless
values of shear stress for each shape combination collapse on to one
another to a single value. Furthermore, for the case of Newtonian
fluid flow, effects of nondimensionalization techniques using various
length scales have been analyzed. It has been determined that through
the use of the square root of cross sectional area for the
characteristic length, as opposed the hydraulic diameter, shear stress
values indicated an increase in accuracy while minimizing the effect
of varying duct shapes.
Despite the research that has been completed for Newtonian fluids,
little attention has been made on modeling NonNewtonian fluids in
similar situations. Some researchers have devoted time and effort in
the analysis of NonNewtonian fluids and their reaction to independent
cross sectional shapes. There has not, however, been an analysis
performed in which the various geometric shapes are compared in their
nondimensional form.
Most text books commonly include tabulated and/or graphical data
describing the most common shapes, thus there is a desire in industry
to have a more robust model that can predict system characteristics as
simply as possible.
The goal of conducting this research is to condense the data that has
been obtained from a number of sources into a single equation that can
be used to describe nonNewtonian flow for a variety of cross
sectional areas. This equation will be used for all shapes where the
only term that is altered is the geometric coefficient relating the
shape of interest to that of fundamental geometries.
 CRISTIAN ENACHE, Département de mathématiques et de statistique, Cité
universitaire Québec (Québec), G1K 7P4, Canada
Some maximum principles for a class of elliptic boundary problems

The subject of this talk will be about a general class of ellipic
boundary value problem on convex bounded domain with a sufficiently
smooth boundary. For a functional combination of the solution and its
gradient we will construct an elliptic inequality for which we can
apply the Hopf maximum principles to derive a priori bounds for
some quantities related to our solution without any explicit knowledge
of the solution itself. We will also give some applications for
geometrical or physical problems.
 KRISTI HOLLOWAY, Memorial University, St. John's, NL
Numerical simulations of viscous fingering involving a single fluid

We present results of numerical simulations of the flow of a single
fluid with a temperature dependent viscosity in the gap of a radial
HeleShaw cell. The simulations are performed using the CFD software,
Fluent. The cell, consisting of two parallel plates separated by a
small gap, is modelled using a grid of 364520 wedge cells. The walls
of the cell are maintained at a constant cold temperature and the cell
is initially filled with cold glycerin. Hot glycerine enters the cell
at a constant rate through a hole in the bottom plate of the cell. A
fingering instability can occur via which the `hot' glycerine forms
fingers that penetrate into the cold glycerine. We present results for
the temperature and velocity profiles of the flow and a stability
diagram which shows the initial parameters for which the flow is
unstable. We also compare our numerical results with experiment.
 RANIS IBRAGIMOV, Department of Applied Mathematics, University of Waterloo,
Waterloo, ON N2L 3G1, Canada
Generation of Internal Tides by an Oscillating Background
Flow Along a Corrugated Slope

The process of internal wave generation by the interaction of an
oscillatory background flow ( U_{0} cos(w_{0} t), V_{0}sin(w_{0} t),
W_{0} sin(w_{0} t) ) over
threedimensional bottom topography is investigated. The topography
considered is a uniform slope with a superimposed corrugation running
directly up and down the slope. Here we present analytical and
numerical analysis of internal tide generation in a stratified,
rotating fluid of infinite depth to better understand the energetics
of the wave generation process. In our model, energy is radiated away
from the bottom as internal gravity waves. Attention is primarily
directed to estimating the flux of energy into the internal wave
field. Since waves are generated not only at the fundamental frequency
w_{0}, but also at all of its harmonics, the energy flux for
both low and high frequency waves is considered. It is shown that the
acoustic limit approximation (the limiting case in which the tidal
excursion, U_{0}/w_{0}, is much less than the scale of the
topography) gives a reasonable approximation for the behaviour of the
energy as the function of the slope, inertial frequency and the basic
flow. The nature of water parcel paths over the corrugations is also
considered.
 BOUALEM KHOUIDER, University of Victoria, Math. and Stat., 3800 Finnerty Road
(Ring Road) Victoria, BC V8P 5C2, Canada
A numerical model for the barotropicbaroclinic interactions
of equatorial waves

We present a high resolution numerical model for the
barotropicbaroclinic interactions of the equatorial waves in a
channel which essentially preserves energy and geostrophic steady
states. A nonlinear incompressible 2d flow and a linear sallow water
system are obtained through Galerkin projection of the nonlinear
betaplane primitive equations on the barotropic and the first
baroclinic modes. The two systems exchange energy through non trivial
interaction terms though total energy is conserved. A high resolution
conservative scheme which preserves geostrophic steady states is
used for each piece: the quasisteady wave propagation algorithm of
R. LeVeque for the baroclinic mode and the central incompressible
scheme of LevyTadmor for the barotropic mode. The interaction terms
are gathered in a single second order accurate scheme to minimize
energy leakage. Validation tests and barotropicbaroclinic wave
interaction runs will be presented.
Joint work with A. Majda.
 GREG LEWIS, University of Ontario Institute of Technology, 2000 Simcoe
Street North, Oshawa, Ontario L1H 7K4
Bifurcations in a differentially heated rotating spherical shell

We study the steady axisymmetric bifurcations that occur in a model
that uses the NavierStokes equations in the Boussinesq approximation
to describe the fluid flow in a differentially heated rotating
spherical shell with radial gravity (i.e., a simple model for
largescale atmospheric dynamics).
The solutions and the corresponding eigenvalues are approximated
numerically from the large sparse systems that result from the
discretization of the partial differential model equations.
 COLIN MACDONALD, Simon Fraser University, Burnaby, BC
Constructing HighOrder RungeKutta Methods with Embedded
StrongStabilityPreserving Pairs

This talk will deal with the construction of fifthorder RungeKutta
schemes with embedded thirdorder strongstabilitypreserving (SSP)
RungeKutta pairs. The original motivation for such pairs was to
evolve Weighted Essentially NonOscillatory (WENO) spatial
discretizations of hyperbolic conservation laws. The thirdorder SSP
scheme would be used near shocks or discontinuities where the SSP
property is useful for minimizing spurious oscillations. Also, WENO
discretizations provide at most thirdorder in space near such
nonsmooth features. The fifthorder scheme would then be used in
smoother regions where WENO provides fifthorder in space and SSP
properties are not necessary.
I will concentrate on the specific techniques used for the
construction of these RungeKutta pairs. I will comment briefly on
the effectiveness of these embedded pairs for evolving hyperbolic
conservation laws, however, these techniques are quite general and
could easily be applied in other areas of study. Thus, I will finish
by noting some other possible applications of these schemes and the
techniques used in their construction.
 FRANKLIN MENDIVIL, Acadia University, Wolfville, NS B4P 2R6
Chaos Games for Wavelet Analysis and Wavelet Synthesis

We start by discussing the classical "Chaos Game" for IFS fractals
and a modification to render selfsimilar L^{2} functions. We then
review the modification of this chaos game to produce an approximation
of a wavelet. By suitably mixing chaos games for different
translations and dilations of the mother wavelet, we can
construct a chaos game to generate approximations to arbitrary L^{2}
functions. This process can also be reversed, yielding a chaos game
for generating the wavelet expansion coefficients of an L^{2}
function.
 SUSAN MOLLOY, Memorial University
Uncertainty Analysis of Powering Prediction Methods of Podded
Propulsor Models

Podded propulsion is a recent innovation in ship powering. The rudder
and propeller configuration raises new challenges in the extrapolation
of model powering data to fullscale values. Traditional testing and
prediction methods have proven unsatisfactory at capturing the
complexity of the interaction between the propeller and pod. Using a
series of model pods, a set of fullscale data and uncertainty
analysis the traditional methods are explored and an alternative
approach to analysing model tests is proposed. A full uncertainty
analysis of the measuring system is included. Using Monte Carlo
Simulation the relative stability of the traditional powering
extrapolation method is compared to the proposed method. The
sensitivity of each method to the powering parameters and correction
factors is determined. Fullscale data is used to evaluate the
results and an acceptable uncertainty range is proposed.
 BENJAMIN ONG, Simon Fraser University
A Moving Mesh Method Based On Level Set Ideas

It has been well documented that mesh adaptation is of critical
importance in the numerical solution of partial differential equations
(PDEs). The accuracy and efficiency improvements often warrant the
extra work entailed in implementing an adaptive mesh.
A promising direction in the field of adaptivity are the socalled
moving mesh methods. In such methods, a mesh equation is solved
(often simultaneously with the original differential equation) for
node velocities, which move/keep the nodes concentrated in regions of
rapid variation of the solution.
In practice, however, most moving mesh methods suffer from an effect
known as "mesh crossing". These typically arise because of some
difference approximations when calculating the mesh velocities.
I will discuss a new moving mesh method based on Level Set ideas.
Such methods eliminate mesh crossings entirely. They also allow for
topology change in the solution or domain. I will then show some
numerical solutions of the Porous Medium Equation (PME) and Fisher's
Equation using this new moving mesh method.
 MATIUR RAHMAN, Department of Engineering Mathematics, Dalhousie University,
Halifax, Nova Scotia B3J 2X4, Canada
Seismic response of earth dams: a theoretical investigation

In order to analyze the safety and stability of an earth dam
during an earthquake, we need to know the response of the dam to
earthquake ground motion so that the inertia forces that will be
generated in the dam by earthquake can be derived. Once the
inertia forces are known, the safety and the stability of the
structure can be determined. In this paper we have investigated an
analytical solution of the seismic response of dams in one
dimension. In real life situation the seismic problem is three
dimensional. However, for an idealized situation, we can
approximate the problem to study in one dimension. The result
obtained in this paper should be treated as a bench mark solution.
 MATEJA SAJNA, University of Ottawa, 585 King Edward, Ottawa, ON
Almost selfcomplementary graphs

A graph X of even order is said to be almost
selfcomplementary if it is isomorphic to the graph obtained from
the complement of X by removing the edges of a 1factor. The study
of almost selfcomplementary graphs was first suggested by Brian
Alspach, who proposed the determination of all possible orders of
almost selfcomplementary circulant graphs. A paper by Dobson and
Sajna, where this problem is solved for a particularly "nice"
subclass of almost selfcomplementary circulants (called
cyclically almost selfcomplementary) shows that the structure of
almost selfcomplementary graphs is much more complex than that of
selfcomplementary graphs.
In this talk, we present some basic results on almost
selfcomplementary graphs and define three binary operations on
graphs that can be used to construct infinite families of almost
selfcomplementary graphs with certain desired properties. One of
these constructions, for example, shows that regular almost
selfcomplementary graphs exist for all even ordersin contrast
with regular selfcomplementary graphs, which exist only for orders
congruent to 1 modulo 4.
This is joint work with Primoz Potocnik.
 SAMUEL SHEN, University of Alberta, Edmonton
The Mathematics in the IPCC Report on Detection, Prediction
and Uncertainty of Climate Changes

This talk will explain some mathematics behind the science in the 2001
Report of the Intergovernmental Panel on Climate Change (IPCC) in the
areas of detection, prediction, and uncertainties of climate change.
Detection is to discern the signals driven by external forces, such as
volcanic aerosols, from the observed climate data. Prediction is the
use of numerical models in forecasting the climate conditions over a
period of several decades with annual or monthly resolution. The
uncertainties exist in observed data, data analysis methods, model
parameters, and statistical inference, and consequently in conclusions
regarding global and regional climate changes. The talk will describe
the IPCC's methods and results of detection, optimal analysis of
observed data, and predictions. But questions remain:
 Has detection taken into account the nonlinear interaction
among the effects of greenhouse gases, volcanic aerosols, and solar
irradiance?
 Have the optimization techniques taken nonstationarity into
account?
 Do we have a satisfactory assessment of the errors of the
observed data over the globe?
References
 [1]

C. K. Folland, N. A. Rayner, S. J. Brown, T. M. Smith, S. S. P. Shen,
D. E. Parker, I. Macadam, P. D. Jones, R. Jones and N. Nicholls,
Global temperature change and its uncertainties since 1861.
Geophys. Res. Lett. 28(2001), 26212624.
 [2]

IPCC Report. Cambridge University Press, 2001, Chs. 2, 9, 10 and
12.
 [3]

C. E. Forest, P. E. Stone, A. P. Sokolov, M. R. Allen and
M. D. Webster,
Quantifying uncertainties in climate system properties with the
use of recent climate observations.
Science 295(2002), 113117.
 [4]

S. S. P. Shen, T. M. Smith, C. F. Ropelewski and R. E. Livezey,
An optimal regional averaging method with error estimates and a
test using tropical Pacific SST data.
J. Climate 11(1998), 23402350.
 MARINA V. TVALAVADZE, Department of Mathematics and Statistics, Memorial University
of Newfoundland, St. John's, NL, Canada
Classification of simple decompositions of Jordan algebras
and superalgebras

This research is dedicated to the classification of simple
decompositions of Jordan algebras and superalgebras. By simple
decomposition we understand representation of an (super)algebra as the
vector sum of two proper simple subalgebras. Based on the explicit
classification of simple Jordan algebras and superalgebras, it was
shown that simple Jordan algebras of type H(F_{n}) and superalgebras
of types osp(n,m), P(n), Q(n), K_{3} and D_{t} have no simple
decompositions. Conversely, all simple decompositions of
(super)algebras of all other types were found in the explicit matrix
form.
 TEYMURAZ TVALAVADZE, Department of Mathematics and Statistics, Memorial University
of Newfoundland, St. John's, NL, Canada
Simple decompositions of Lie superalgebras

This talk will report on the recent classification of simple
decompositions of Lie superalgebras. The similar problem is
wellstudied in the case of Lie algebras. Generally speaking, we
consider the representation of the given simple Lie superalgebra as
the sum of two proper simple subalgebras. In this research the
classification of simple Lie algebras and superalgebras play a key
role. In particular, it was shown that classification of simple
complex Lie algebras can be extended to the case of algebraically
closed field of zero characteristic. Also, all simple decompositions
of classical Lie superalgebras were explicitly classified and
represented in the matrix form.
 NAVEEN VAIDYA, York University, 4700 Keele St., Toronto, ON M3J 1P3
Grownin Defects Modeling of InSb Crystals

In this talk, we will present a model for grownin point defects
inside indium antimonide crystals grown by the Czochralski
technique. This model includes the Fickian diffusion and recombination
mechanism. This type of model is used for the first time to analyse
grownin point defects in indium antimonide crystals.
The temperature solution and the advance of the meltcrystal interface
are based on a recently derived perturbation model. We study the
effect of thermal flux on the point defect patterns during and at the
end of the growth process. Our results show that the concentration of
excessive point defects is positively correlated to the heat flux in
the system. Based on the result of our twodimensional model, we can
conclude that for growing lessdefect crystals of larger radius one
has to reduce the heat transfer coefficient by a suitable amount from
the lateral surface of the crystal.
 MICHAEL WAITE, McGill University, Department of Atmospheric and Oceanic
Sciences, 805 Sherbrooke St. W., Montreal, QC H3A 2K6
Numerical simulations of vortical and wave motion in stably
stratified turbulence

Turbulence in a stratified fluid can be decomposed into a potential
vorticitycarrying (vortical) component and an internal wave
component. In the limit of strong stratification, internal waves
become weakly nonlinear while vortical motion remains fully nonlinear;
the two components have very different limiting dynamics.
In this talk, we contrast stratified turbulence dominated by vortical
and wave motion. Numerical simulations are presented of a Boussinesq
fluid, in which largescale vortical and wave motion are forced
separately over a wide range of stratifications. With vortical forcing
and strong stratification, energy spectra are found to be quite
different from observations in the atmosphere and ocean.
However, we argue that these results are consistent with the limiting
dynamics, which for vortical motion predict decoupled layers of
twodimensional turbulence. We show that vertical decoupling breaks
down at a scale of U/N where U is the RMS velocity and N is the
BruntVäisälä frequency, in agreement with theory. These
findings are contrasted with the waveforced simulations.
 STEVEN QIANG WANG, School of Mathematics and Statistics, Carleton University,
Ottawa, Ontario K1S 5B6, Canada
Generalized Inflations and Null Extensions

An inflation of an algebra is formed by adding a set of new
elements to each element in the original or base algebra, with the
stipulation that in forming products each new element behaves
exactly like the element in the base algebra to which it is
attached. Clarke and Monzo have defined the generalized inflation
of a semigroup, in which a set of new elements is again added to
each base element, but where the new elements are allowed to act
like different elements of the base, depending on the context in
which they are used. Such generalized inflations of semigroups are
closely related to both inflations and null extensions. Clarke and
Monzo proved that, for a semigroup base algebra which is a union of
groups, any semigroup null extension must be a generalized
inflation, so that the concepts of null extension and generalized
inflation coincide in the case of unions of groups. As a
consequence, the collection of all associative generalized
inflations formed from algebras in a variety of unions of groups
also forms a variety.
In this paper we define the concept of a generalized inflation for
any type of algebra. In particular, we allow for generalized
inflations of semigroups which are no longer semigroups
themselves. After some general results about such generalized
inflations, we characterize for several varieties of bands which
null extensions of algebras in the variety are generalized
inflations, and which of these are associative. These
characterizations are used to produce examples which answer, in
our more general setting, several of the open questions posed by
Clarke and Monzo.
This is a joint work with Shelly L. Wismath.
 RONG WANG, Dalhousie University, Halifax, NS
A numerical study of time integration schemes for the cubic
Schrödinger equation

The methodoflines (MOLs) is a popular approach for the computational
treatment of partial differential equations. We describe two software
packages which use collocation with a Bspline basis for the spatial
discretization, and which use, for the resulting differential
algebraic equations (DAEs), either DASSL or RADAU5. The former is
based on a family of backward differentiation formulas (BDFs), while
the latter uses a fifthorder Astable implicit RungeKutta
scheme. These MOL packages use a sophisticated algorithm for spatial
adaptation and spatial error control. We apply the software to the
nonlinear cubic Schrödinger equation in one space
variable. Because this problem has all the eigenvalues on the
imaginary axis, Astablity is necessary for the time integration
schemes. Only first and second order BDF methods are
Astable. Computational results show that while DASSL is unable to
obtain a correct solution unless we restrict the maximum order of the
method to 2, RADAU5 handles the problem without difficulty.
 JIN YUE, Dalhousie University
The 1856 Cayley Lemma Revisited

Arthur Cayley (18211895), an eminent English mathematician of the
19th century, made many significant contributions to a number of areas
of mathematics, including algebra, invariant theory, projective
geometry and group theory. Among his most remarkable works may be
mentioned the "ten memoirs on quantics", commenced in 1854 and
completed in 1878.
We will review one of the works in this series, namely "A second
memoir on quantics" (Phil. Trans. Roy. Soc. London
146(1856), 101126) that contains a result known in classical
invariant theory as "Cayley's Lemma". It concerns the action of
SL(2) on the space of homogeneous polynomials of degree n. We
will then formulate and prove an analogue of Cayley's Lemma in the
invariant theory of Killing tensors of pseudoRiemannian geometry.

