


Communications libres
Org: C. Robert Miers (Victoria) [PDF]
 LENNARD BAKKER, Brigham Young University
The Multiplier Group of a Quasiperiodic Flow
[PDF] 
The multiplier group of an Falgebraic quasiperiodic flow on the
ntorus is shown to be a finite index subgroup of the group of units
in the ring of integers of the real algebraic number field F. It is
also shown that the multiplier group of a quasiperiodic flow being
{1,1} implies the that the quasiperiodic flow is transcendental.
The converses of these are validated for quasiperiodic flows on the
2torus.
 ROBB FRY, Thompson Rivers University, Kamloops, BC
Beyond partitions of unity
[PDF] 
The tried and true method of uniformly approximating continuous
functions by smooth functions on Banach spaces is through the use of
smooth partitions of unity. Much effort over the last several decades
has gone into establishing the existence of such smooth partitions of
unity on ever wider classes of nonseparable Banach spaces. One
weakness of this technique is that it is difficult to arrange for the
approximating function to have stronger properties in addition to
C^{p}Fréchet smoothness.
In this talk we shall first review the method of approximation via
partitions of unity, and then discuss more recent approaches in which
the smooth approximating function can be chosen also to be Lipschitz
or possess a uniformly continuous derivative.
 MUDASSAR IMRAN, Arizona State University, PO Box 871804, Tempe, AZ
852871804, USA
The Model of Antibiotic Resistance in Biofilm
[PDF] 
A mathematical model of the effect of antibiotics on bacterial biofilm
is considered. According to the US National Institute of Health,
"Biofilms are medically important, accounting for over 80% of
microbial infections in the body", such as otitis media, the most
common acute ear infection in children in the US. Biofilms are highly
resistant to antibiotics. Consequently, very high and/or longterm
doses are often required to eradicate biofilmrelated infections. We
consider a twocompartments, chemostatbased model where one
compartment has a very high dilution rate as compared to the other
compartment. The high dilution rate compartment represents the fluid
environment and the low dilution rate compartment represents the
stagnant biofilm environment. A constant supply of nutrient and a
periodically fluctuating antibiotic agent is supplied to the high
dilution compartment. The model assumes that antibiotic increases the
death rate as its concentration is increased. We use persistence as
well as global bifurcation results for a mathematical analysis of
periodic solutions. The model consists of a system of nonautonomous
differential equations which govern the dynamics of the bacteria in
biofilm.
This is joint work with Hal Smith.
 INJAE KIM, University of Victoria
Smith Normal Form and Acyclic Matrices
[PDF] 
An approach, based on the Smith Normal Form, is introduced to study
the spectra of symmetric matrices with a given graph. The approach
serves well to explain how the path cover number (resp. diameter of a
tree T) is related to the maximum multiplicity occurring for an
eigenvalue of a symmetric matrix whose graph is T (resp. the
minimum number q(T) of distinct eigenvalues over the symmetric
matrices whose graphs are T). The approach is also applied to a
more general class of connected graphs G, not necessarily trees, in
order to establish a lower bound on q(G).
 MAIA LESOSKY, Guelph
Medical Imaging and Euclidean Motion Deconvolution
[PDF] 
This talk will discuss the application of deconvolution methods to
medical imaging data. The Radon transform forms the backbone for most
computerized tomography (CT) imaging systems. Reconstruction of these
images requires an implementation of the inverse Radon transform.
There are a number of inversion algorithms available, however, most
have serious deficiencies. This approach formulates the Radon
transform as a convolution integral over the Euclidean motion group.
 XIAO PING LIU, Department of Mathematics and Statistics, University of
Regina, Regina, SK S4S 0A2
Determinantal inequalities for certain classes of totally
nonnegative matrices
[PDF] 
An nbyn matrix A is called totally nonnegative, TN (totally
positive, TP) if the determinant of every square submatrix
(i.e., minor) of A is nonnegative (positive). A
collection S of bounded ratios are said to be generators for all
such bounded ratios if any bounded ratios can be written as a product
ratios from this collection S. We define a particular class of
totally nonnegative matrices, called STEP1, and establish all the
generators for STEP1 matrices. All such generators have been shown to
be bounded ratios for general TN matrices.
 TUFAIL MALIK, Arizona State University, PO Box 871804, Tempe, AZ 852871804,
USA
A ResourceBased Model of Microbial Quiescence
[PDF] 
To analyze the ecological features of microbial quiescence, a model is
proposed that involves "wakeup" rate and "sleep" rate at which
the population transitions from a quiescent to an active state and
back, respectively. These rates depend continuously on the resources
and turn on and off at resource thresholds which may not coincide.
The usual dichotomy is observed: the population is washed out under
environmental stress and a single "survival" steady state exists
otherwise. Proportional nutrient enrichment is used to explore
analytically as well as numerically the nature of the steady state
which bifurcates from the washout state.
This is joint work with Hal Smith.
 MITJA MASTNAK, Dept. of Math, UBC, Room 121, 1984 Mathematics Road,
Vancouver, BC V6T 1Z2
About Linear Spaces of Matrices
[PDF] 
If L is an mdimensional linear subspace of M_{n×p}, the
space of n×p matrices, then we can identify the embedding k^{m}[( @ )  (® )] L Í M_{n×p} with a bilinear map
k^{m} ×k^{n} ® k^{p} or with a linear map k^{m}Äk^{n}® k^{p}.
If we "switch" k^{m} and k^{n} then we get an embedding k^{n}®M_{m×o}, and hence an ndimensional linear subspace L¢ of
M_{m×o}. We study the relationship between L and L¢ and
give examples of situations where this duality can be exploited.
 MICHAEL SZAFRON, Department of Mathematics and Statistics, University of
Saskatchewan, Saskatoon, SK
The Probability of Knotting after a Local Strand Passage in
an Unknotted SAP
[PDF] 
Due to DNA's structure, it is prone to several different topological
entanglement problems. The cell must be able to resolve each of these
problems because the problems interfere with vital cell functions.
For instance, knotted DNA cannot replicate successfully. The cell
therefore must have some mechanism by which the DNA can be unknotted.
This mechanism is the interaction of the DNA with the topoisomerase
enzymes.
The topoisomerase enzymes interact locally with the DNA and pass one
strand of DNA through itself via the enzymebridged transient break in
the DNA [RW94]. Since these local strandpassages can potentially
change the knottype of the DNA [DSKC85], [WC91], experimentalists
can use the frequency of knots produced to characterize topoisomerase
action on DNA topology [WC86].
It is of interest whether these local strandpassages are implemented
at random locations in the DNA. In order to investigate this problem,
a simplified model of an unknotted ring polymer was implemented via
Monte Carlo simulation to estimate the probability that a ring polymer
has knottype K after a local strand passage has occurred within the
ring polymer. The model, some estimates of the knotting
probabilities, and an observation will be presented.
 LUIS VERDESTAR, Universidad Autonoma Metropolitana, Mexico City
Computation of the matrix exponential via the dynamic
solution
[PDF] 
Let A be a square matrix with characteristic (or minimal) polynomial
w(x), of degree n+1. The dynamical solution g(t) associated with
w is the solution of the homogeneous differential equation
w(D)y(t) = 0 that has the initial values D^{k} g(0) = 0 for 0 £ k < n and D^{n} g(0) = 1.
The matrix exponential is given by
e^{tA} = 
n å
k=0

D^{k} g(t) w_{nk}(t), 

where the w_{j} are the Horner polynomials associated with w. We
explore some analytic and numerical aspects of the use of the above
formula, and try to explain the sources of the main computational
problems. Other functions of matrices can be computed by analogous
formulas. See L. VerdeStar, Functions of matrices, Linear
Alg. Appl. 406(2005), 285300.

