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Contributed Papers Session / Communications libres
(Stanley Kochman, Organizer)

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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 pseudo-monotone 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, complex-valued, 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 Leaf-Preserving Bijection

A number of combinatorial objects-labeled trees, allowable pairs of input-output permutations for priority queues, factorizations of an n-cycle into transpositions, and parking functions-are all enumerated by the same formula: (n+1)^n-1. A series of bijections-many of them related-have 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
Set-theoretic 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 set-theoretic 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 time-varying 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 uniqueness-compactness 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 G₁ and G₂, the bipartite rainbow ramsey number BRR(G₁, G₂) 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 G₁ or a rainbow copy of G₂. We show that BRR(G₁, G₂) exists if and only if G₁ is a star or G₂ is a star forest. Exact values and bounds for BRR(G₁, G₂) for various pairs of graphs G₁ and G₂, 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 max-type difference equation x[n+1] = max_{1/x[n], A[n]/x[n-1]}

Throughout this talk we will discover how the periodicity and boundedness nature of the positive solutions of the above MAX-TYPE 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 tradition-bound 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 well-known 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  316-8511, 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 N₀ 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 infinite-dimensional, 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 Hodgkin-Huxley 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 Cⁿ. 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, Ciocan-Fontanine 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.