




Contributed Papers Session / Communications libres (Stanley Kochman, Organizer)
 MONICA COJOCARU, Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario K7L 3N6
Dynamical study of solutions to variational inequalities and
complementarity problems

(Joint work with George Isac)
Variational Inequalities (VIP) and Complementarity Problems (CP) are
the mathematical tools used in the study of equilibrium, in the
physical or economical sense. The dynamical study of solutions to VIP
or CP is interesting especially from the applications point of view, in
practical problems. In this paper, by using differential inclusions, we
will show that solutions to VIP or CP defined by pseudomonotone
operators have interesting stability properties.
 MAHMOUD FILALI, University of Oulu and University of Western Ontario
On minimal idempotents in LUC(G)^{*} and in similar algebras

Let G be a locally compact group, and let LUC(G) be the space of
bounded, complexvalued, uniformly continuous functions with respect to
the right uniformity of G. As is well known the Banach space dual
LUC(G)^{*} is a Banach algebra with the (first) Arcus product. We
give examples of minimal idempotents in this algebra and in similar
algebras. We study the ideals generated by these idempotents.
 ANGELE HAMEL, Wilfred Laurier University, Waterloo, Ontario N2L 3C5
Allowable Pairs and Labeled Trees: A LeafPreserving Bijection

A number of combinatorial objectslabeled trees, allowable pairs of
inputoutput permutations for priority queues, factorizations of an
ncycle into transpositions, and parking functionsare all
enumerated by the same formula: (n+1)^{n1}. A series of
bijectionsmany of them relatedhave been constructed between two
or more of these. Here we introduce and prove a direct bijection
between allowable pairs and labeled trees that has an additional
property not present in previous direct bijections: our bijection maps
increasing sequences in the output permutation of the allowable pair to
leaves in the tree. This gives us a full understanding of the
underlying tree structure of allowable pairs. For instance, we can use
this understanding to construct the analogue of a Prufer code for
allowable pairs. New enumerative applications can be obtained in this
way.
 MOSES KLEIN, Lawrence University, Appleton, Wisconsin 54912, USA
Settheoretic perspectives on subgroups of Z^{w}

It has long been known, since work of Baer (1937) and Specker (1950)
that the group additive group Z^{w} of infinite integer
sequences is not a free abelian group but has the property that all of
its countable subgroups are. More recent work has partially
characterized the subgroups that can be embedded in
Z^{w}, including some results provably independent of ZFC
set theory. This paper will summarize and extend these results, with
emphasis on identifying the settheoretic principles needed to settle
the independent questions.
 WEIJIU LIU, University of Cincinnati, Cincinnati, Ohio 45221, USA
Stabilization, controllability and
observability for electromagnetic fields in
dissipative material regions

We consider the problem of stability, observability and controllability
for timevarying electromagnetic fields in dissipative material regions
which are described by Maxwell's equations. In this model, the
interaction between the electric and magnetic fields and the media is
taken into account and therefore the conductivity and polarization is
introduced. The conductivity is due to the application of the external
electric field to a conducting material while the polarization arises
from the application of the external electric field to n dielectic
materials, each of which has its own dynamics governed by an ordinary
differential equation. By using the Lyapunov method, we show the
fields are exponentially stable. With the help of multiplier technique
and uniquenesscompactness argument, we prove that the fields are
observable, that is, the initial states of the fields can be recovered
from the observation of their magnetic intensity outputs on the
boundary of a region during a period of T. Via the Lions' Hilbert
uniqueness method, we establish the exact controllability for the
fields, that is, currents can be specified on the boundary of a region
so that a desired and specified electromagnetic field is produced
within the region.
 ORTRUD R. OELLERMANN, University of Winnipeg, Winnipeg, Manitoba
Bipartite rainbow Ramsey numbers

Let G and H be graphs. Suppose that the edges of H are coloured
with any number of colours. Then the resulting graph is said to
contain a monochromatic (rainbow) copy of G if it contains a subgraph
isomorphic to G, all of whose edges are coloured with the same colour
(no two of which are coloured with the same colour, respectively). For
bipartite graphs G_{1} and G_{2}, the bipartite rainbow ramsey number
BRR(G_{1}, G_{2}) is the smallest N such that if the edges of
K_{N,N} are coloured with any number of colours, then the resulting
colouring contains either a monochromatic copy of G_{1} or a rainbow
copy of G_{2}. We show that BRR(G_{1}, G_{2}) exists if and only if
G_{1} is a star or G_{2} is a star forest. Exact values and bounds for
BRR(G_{1}, G_{2}) for various pairs of graphs G_{1} and G_{2}, for which
the bipartite rainbow ramsey number is defined are established.
(joint work with L. Eroh)
 MICHAEL RADIN, Department of Mathematics and Statistics,
Rochester Institute of Technology, Rochester, New York 14623,
U.S.A.
Boundedness and periodicity nature of positive
solutions of a maxtype difference equation x[n+1] = max_{1/x[n], A[n]/x[n1]}

Throughout this talk we will discover how the periodicity and
boundedness nature of the positive solutions of the above MAXTYPE
DIFFERENCE EQUATION depends on the relationship of the parameters
A[n], not the initial conditions.
(1) Give brief history of the problem and how it relates to the 3x+1
CONJECTURE.
(2) Discuss the Autonomous Case where A[n] is a constant sequence
and the periodicy nature of positive solutions.
(3) Discuss the Case where A[n] is a prime period 2 sequence and how
the periodicty nature relates to the constant case in (2).
(4) Discuss the Case where A[n] is a prime period 3 sequence; we
will discover drastic changes in the periodicity nature of solutions
and for the first time discover unbounded solutions.
(5) Give some examples where a certain period appears and where
unbounded solutions appear.
(6) Conclude by discussing some generalizations as the period of
A[n] increases.
 DIETER RUOFF, University of Regina, Regina, Saskatchewan S4S 0A2
An inventory of topics which belong to elementary
hyperbolic geometry

What is presented in a course or in a textbook on elementary hyperbolic
geometry could be termed a loosely connected sequence of topics.
Typically included are statements concerning the distance between and
common perpendiculars of two lines, improper triangles, the angle of
parallelism, the theory of area and possibly the connection between a
Lambert quadrilateral and a right triangle. Surprisingly this list does
not so much reflect a traditionbound selection of topics as it
represents fairly closely the state of knowledge of the field. The aim
of the talk is to identify a number of theorems which mostly have
wellknown counterparts on the Euclidean side but which have remained
unnoticed or unproved to this day. The list contains statements about
transversals of a pair of lines, parallelograms, proportionality and
curves related to the circle.
 NOBUHISA SAKAKIBARA, Faculty of Engineering, Ibaraki University, Hitachi 3168511,
Japan
Stieltjes moment problems on abelian *semigroups

(This is a joint work with T. M. Bisgaard (Denmark).)
An abelian *semigroup S is perfect (resp. Stieltjes
perfect) if every positive definite (resp. completely positive
definite) function on S admits a unique disintegration as an integral
of characters, that is, hermitian multiplicative functions (resp.
nonnegative characters). We prove that every Stieltjes perfect
semigroup is perfect. The converse was known earlier for semigroups
with neutral element, but is here shown to be not true in general.
Furthermore, we prove that an abelian *semigroup is perfect if for
each s Î S there exist t Î S and m,n Î N_{0} such that
m+n ³ 2 and s+s^{*}=s^{*}+mt+nt^{*}. This was known earlier only with
the equality replaced by s=mt+nt^{*}. The equality cannot be replaced
by s+s^{*}+s=s+s^{*}+mt+nt^{*} in general, but for semigroups with neutral
element it can be replaced by s+p(s+s^{*})=p(s+s^{*})+mt+nt^{*} for an
arbitrary p Î N.
 GIOVANNI SAMAEY, KULeuven, Department of Computer Science, Celestijnenlaan 200A,
3001 Leuven, Belgium
Numerical computation of connnecting orbits in delay
differential equations

We discuss the numerical computation of homoclinic and heteroclinic
orbits in delay differential equations. Such connecting orbits are
approximated using projection boundary conditions, which involve the
stable and unstable manifolds of a steady state solution. The stable
manifold of a steady state solution of a delay differential equation
(DDE) is infinitedimensional, a problem which we circumvent by
reformulating the end conditions using a special bilinear form. The
resulting boundary value problem is solved using a collocation method.
We demonstrate results, showing homoclinic orbits in a model for neural
activity and travelling wave solutions to the delayed HodgkinHuxley
equation. Our numerical tests indicate convergence behaviour that
corresponds to known theoretical results for ODEs and periodic boundary
value problems for DDEs.
 BOGDAN SUCEAVA, Michigan State University
Riemannian obstructions to minimal isometric immersions in Euclidean
spaces

The classical obstruction to minimal isometric immersions into
Euclidean spaces is Ric ³ 0. In this talk, by applying Chen's
invariants, we present a method to construct explicit examples of
Riemannian manifolds with Ric < 0 which don't admit any minimal
isometric immersion into Euclidean spaces for any codimension.
 ALEXANDER YONG, University of Michigan, Ann Arbor, Michigan, USA
On Quantum cohomology for the Grassmannian

Let X be the complex Grassmannian manifold whose points are k
dimensional subspaces of C^{n}. We prove a theorem that
describes a property of the multiplication rule for two Schubert
classes in the (small) quantum cohomology ring of X, due to Bertram,
CiocanFontanine and Fulton. We complete an essentially combinatorial
proof of a recent result of Fulton and Woodward that characterizes the
smallest degree of q that appears in such a product. The original
proof had used [`(M)]_{0,n}(X,d), the space of degree d
stable maps of genus 0 curves with n marked points to X. No
background in Schubert calculus is required.

