Communications libres
Org: Tatyana Foth (Western) [PDF]
 AMAL AMLEH, Saint Mary's University
On SecondOrder Rational Difference Equations
[PDF] 
In this talk we present a summary of recent and new results on the global character of solutions of the secondorder rational difference equation
x_{n+1} = 
a+ bx_{n} + gx_{n1}
A + Bx_{n} + Cx_{n1}

, n = 0,1,... 

with nonnegative parameters a, b, g, A, B, C and with
arbitrary nonnegative initial conditions x_{1}, x_{0} such that the
denominator is always positive. Some extensions to higherorder
rational difference equations will also be presented.
 MICHAEL CAVERS, University of Regina, Department of Mathematics and Statistics, Regina, SK S4S 0A2, Canada
Reducible inertially arbitrary matrix patterns
[PDF] 
An n by n nonzero (resp. sign) pattern A is a matrix
with entries in {*,0} (resp. {+,,0}). The inertia of a matrix
A is the ordered triple (a_{1},a_{2},a_{3}) of nonnegative integers where
a_{1} (resp. a_{2} and a_{3}) is the number of eigenvalues of A with
positive (resp. negative and zero) real part. A is
inertially arbitrary if each nonnegative integer triple (a_{1},a_{2},a_{3})
with a_{1}+a_{2}+a_{3}=n is the inertia of a matrix with nonzero (resp. sign) pattern A. Some observations regarding which inertias
A and B may allow to guarantee A ÅB is inertially arbitrary are presented. It is shown that there exists noninertiallyarbitrary nonzero (resp. sign) patterns A and B such that A ÅB is inertially arbitary.
 VIRGINIE CHARETTE, Université de Sherbrooke, Sherbrooke, QC
Affine deformations of the holonomy group of a threeholed sphere
[PDF] 
Let T be a complete hyperbolic surface homeomorphic to a threeholed
sphere and let G denote the image under the holonomy representation
of its fundamental group. Identifying the group of hyperbolic
isometries with an appropriate component of the group of isometries of
Minkowski spacetime, we may consider affine deformations of G; we may
ask, when does this affine deformation act properly discontinuously on R^{3}? An important invariant for affine isometries with nonelliptic linear part is the Margulis invariant, which is a measure of signed Lorentzian displacement. In this paper, we show that an affine deformation of G acts properly discontinuously if and only if the Margulis invariant is positive for each of the three isometries
corresponding to the pant holes of T. More precisely, we show that
such an affine deformation admits a fundamental domain.
This is joint work with Drumm and Goldman.
 ALEXEI F. CHEVIAKOV, University of British Columbia, Vancouver, BC
Construction and Applications of Nonlocally Related Systems of Partial Differential Equations
[PDF] 
For a given Partial Differential Equation (PDE) or a system of PDEs,
with n ³ 2 independent variables, I will describe a systematic
(and rather general) framework to find PDE systems nonlocally
related to the given one. Such nonlocally related PDE systems have solution sets that are equivalent to the solution set of the given system (i.e., any solution of a nonlocally related system yields a solution of the given system, and, conversely, any solution of the given system yields a solution of the nonlocally related system). Moreover, the solution of any boundary value problem posed for the given PDE system is embedded in the solution of a boundary value problem posed for the nonlocally related (potential) system (and the converse also holds).
Due to nonlocal relations and the equivalence of solution sets, any
general method of analysis (symmetry, conservation law, qualitative,
perturbation, numerical, etc.), when applied to one of such nonlocally
related PDE systems, can yield new results. In particular, new
conservation laws, nonlocal symmetries and new exact solutions have
been found for many nonlinear PDE systems arising in applications.
I will discuss several illustrative examples: the Nonlinear Wave
equation, Planar Gas Dynamics equations, equations of Nonlinear
Elastodynamics, and Plasma Equilibrium equations in 3D.
The talk is aimed at the broad audience of applied mathematicians and
researchers working with PDE models.
This is a joint work with George Bluman.
 FRANKLIN MENDIVIL, Math. Dept., Acadia University, Wolfville, NS
Annealing a GA for Constrained Optimization
[PDF] 
We consider the problem of adapting a Genetic Algorithm (GA) to
constrained optimization problems, using a dynamic penalty approach as
a type of annealing. We present two different methods for ensuring
almost sure convergence to a globally optimal (feasible) solution. The
first of these involves modifying the GA operators to yield a
Boltzmanntype distribution while the second incorporates a dynamic
penalty along with a slow annealing of the acceptance probabilities.
 PAUL POTGIETER, University of South Africa, Department of Decision Sciences, PO Box 392, Pretoria, 0003 South Africa
Large fluctuations of complex oscillations
[PDF] 
Recent precise estimates on the frequency of occurrence of intervals
of rapid growth on a Brownian path are discussed. These are then
used, together with the theory of Gaussian algebras, to obtain
similar estimates for the large fluctuations of complex
oscillations, which can be seen as Brownian motion generated by the
set of KolmogorovChaitin complex strings. This yields the Hausdorff
dimension of the rapid points of such complex oscillations.
 FREYDOON RAHBARNIA, Ferdowsi University of Mashhad, Department of Mathematics, PO Box 1159, Mashhad 91775, Iran
Knots, Projections and Graphs
[PDF] 
In this paper, we study graphs related to knots. We investigate the
connection between knot projections and graphs. We use the related
graphs to catalog knot projections.
 YONGJUN XING, University of Regina, 3737 Wascana Parkway, Regina, SK S4S 0A2
On the Spread of Real Symmetric Matrices with Entries in an Interval
[PDF] 
The spread of a matrix has extensive and practical applications in
combinatorial optimization problems and cybernetics problems. There
are many papers on the spread of a symmetric matrix, but restricting
the entries of such n×n symmetric matrices to each lie in [a,b] seems to be a new view of this problem. As a first step, we show that the entries must equal a or b in the case when the spread is maximum. Next, when the spread attains the upper bound of Mirsky's seminal result, we describe the structure of those matrices. Then we focus our study on the maximal value of the spread and the corresponding structure of the matrix that achieves the maximum spread over all real symmetric, n×n matrices, whose entries lie in a given interval. Matlab is used as a tool to aid the verification of some cases.
