CMS-MITACS Joint Conference 2007

We use the theory of Lyapunov coefficients to estimate the number, and characterize the stability, of limit cycles associated with an SIR model that employs saturated incidence function of the general form: g(I) = kI^h/(1+aI^h), where k, a and h are parameters. This study confirms that the model may have at most two limit cycles when h=2. Furthermore, it is shown that for the case h=3, the model may have a maximum of three limit cycles. The stability of these limit cycles is characterized based on the signs of the Lyapunov coefficients.

In this talk, we propose an efficient splitting domain decomposition method (S-DDM) for solving parabolic equations and study a splitting ELLAM for convection-dominated diffusion problems in high-dimensions. We apply the S-DDM scheme by incorporating the upstream methods to simulate the groundwater contaminant transport in porous media.

Herring tag data is available starting from 1936 in British Columbia. We are looking at herring stray rates between five geographically distinct populations defined by the Department of Fisheries and Oceans (DFO). We will create a multi-state population model to estimate yearly stray rates between populations. Based on deviances from these estimates we will identify years where there have been noticeable quantitative changes in herring stray behaviour. Hopefully these changes can be associated with recorded environmental indicators. Special consideration will be given to how to treat missing data resulting from both fishery closures and missed tagging events.

Instantaneous frequency (IF) is a critical parameter in describing non-stationary signals whose frequency characteristics vary over time.

In real application, it is usually difficult to obtain accurate IF estimation of signals in terms of their multi-component feature and the presence of noise. One kind of popular and useful approach for measuring the IF is based on the property of time-frequency representation. In the talk, we give a short review about two basic techniques of them: peak detection method and the reassignment method. And then we extend such technique with the recently developed time-frequency representation: the Stockwell transform. The improvement of our new methodologies are shown by numerical simulations.

With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced bipartite subgraph of G, denoted a₂(G). We prove a strenghtening of this conjecture by showing that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced forest, denoted F(G). Moreover, we characterize the graphs maximizing the average distance among all graphs G having a fixed number of vertices and a fixed value of F(G) or a₂(G). Finally, we conjecture that the average distance between two vertices of a connected graph is at most half the maximum order of an induced linear forest (where a linear forest is a union of chains).

This is joint work with Pierre Hansen (HEC Montréal) and Alain Hertz, Rim Kilani and David Schindl (Ecole Polytechnique de Montréal).

The Spekkens toy model is an interesting example of how to augment classical physics in order to perform several quantum informational tasks using limited resources. We revisit the Spekkens toy model and look at the different representations for the group of operations on a single toy bit. We show that in the representation of the operators as Euler rotations, there exist rotations that obey the knowledge balance principle, yet are not present in Spekkens' original group. We demonstrate that this expanded group of single toy bit operations, which includes Spekkens' original operations as a subgroup, is isomorphic to the extended Clifford group for one qubit (modulo scalar multiples of the identity). We also investigate the case for two toy bits again expanding the group of toy operations to include some, but not all, of the extended operations.

In this talk, we will show that the number M(n,k) of partitions of nonnegative integer n with k parts can be described by a set of [(k)\tilde] polynomials of order k-1 in Q_[(k)\tilde], where [(k)\tilde] denotes the least common multiple of 1,2,...,k and Q_[(k)\tilde] is the quotient of n when divided by [(k)\tilde]. In addition, the sets of the [(k)\tilde] polynomials are obtained explicitly for k=3,4,5, and 6.

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives information about the structure of the group of automorphisms of such algebras.

It was shown recently that the central path can be bent along the simplex path of Klee-Minty cubes. This lead to tightening the iteration complexity bound of central path following interior point methods. Further, intriguing analogs between edge-paths and central paths arise. We conjecture that the order of the largest total curvature of the central path is the number of inequalities, and that the average diameter of a bounded cell of an arrangement is less than the dimension. We substantiate these conjectures and prove a continuous analog of the d-step conjecture.

Joint work with A. Deza, E. Nematollahi and Y. Zinchenko.

The John-Nirenberg inequality characterizes functions in the space BMO in terms of the decay of the distribution function of their oscillations over a cube [JN, 1961]. In joint work with Galia Dafni, we prove a John-Nirenberg type inequality for functions in the space Q_a (Rⁿ), which is a modified version of the conjecture by Essén, Janson, Peng and Xiao [EJPX, 2000]. We construct a function, as a counterexample, to show the necessity for this modification.