Cameron Christou
University of British Columbia
Talk title: Dither and error diffusion in image processing

Abstract:
The sampling theorem tells us that we can perfectly reproduce a continuous (band limited) waveform from a discrete set of sample points. However, the values at these sample points must be truncated or rounded so that the information can be digitally recorded. This is a process called quantization, and it necessarily causes an error in the reproduced signal.

How serious is this quantization error? What can be done to minimize its effect? We will investigate the answers to these questions. Applications to image processing will be considered for illustrative purposes.

Simon Gemmrich
McGill University
Talk title: Boundary integral equations for the Laplace Beltrami operator on the unit sphere

Abstract:
In this talk I will present how the standard indirect boundary element approach can be applied to solve the Dirichlet Problem for the Laplace Beltrami operator on the unit sphere. I will mention a representation formula, the standard boundary integral operators along with their mapping properties and show some numerical results for the single layer equation.

Seungji Ha
University of Alberta
Talk title: Finite amplitude evolution of baroclinic waves

Abstract:
Synoptic scale (~1000km) vortices in the atmosphere and mesoscale (~100km) vortices in the ocean are omnipresent in the general circulation. Much of the initial development of these system can be explained by barocinic instability. I would like to develop a theoretical description of the evolution of the potential vorticity and the amplitude of marginally unstable baroclinic wave in the neighborhood of the minimum critical value of the vertical shear. At minimum critical shear, linear and low-level potential vorticity advection vanishes so that a balance between time evolution, nonlinearity, and dissipation becomes the dominant dynamics.

Jeffrey Haroutunian
University of Calgary
Talk title: A spatial generalization of the Ricker model: break of chaos

Abstract:
The Ricker model x_n+1=x_nexp(r(1-x_n/k)) is a well-studied population model that is known to be chaotic for sufficiently large r. It has been found that the introduction of a small, positive perturbation causes a break of chaos and gives rise to a stable 2-cycle. Such a perturbation plays the role of immigration. Other types of perturbations (random, emigration) have also been studied. We will give a spatial generalization of this model, specifically focusing on two dimensions. We apply the aforementioned perturbations to our model, and discuss whether the results from the standard model carry over to higher spatial dimensions. We will also discuss new phenomena that arise out of our model.

Maria Khomenko
University of British Columbia
Talk title: Instabilities in gravity driven flows under stiff boundary conditions

Abstract:
When a fluid is flowing down an incline its behaviour is unstable and it forms streaks or `fingers'. These instabilities are of interest and one can use linear stability analysis to identify the first mode to become unstable. The mathematical model used for analysis originates from Navier-Stokes equations and lubrication theory. The boundary conditions assume that the fluid is covered by an elastic beam, which makes the problem stiffer than the classical case involving only the fluid surface tension. This behaviour is studied experimentally using corn syrup as the fluid and a latex sheet as the elastic beam. The results provide a realistic approximation to the wave-front behaviour.

Abdalla Mansur
Queen's University
Talk title: The Maslov index and instability of the periodic solutions for the rhombus four body problem

Abstract:
This talk concerns instability of periodic solutions for the rhombus 4-body problem. Instability for these solutions will be descibed using the geometric and analytic techniques rather than the numerical techniques. We will show that the stable and unstable manifolds along the periodic solutions describe a closed curve of G-Lagrangian subspace, to which we can associate a Maslov index defined to be the number of intersection of a closed curve of G-Lagrangian subspace with a fixed G-Lagrangian subspace. In the circular case (the linearized system will be time independent), it will be shown that this index is equivalent to the reduced index and this index is equal to 2. The question of stability then can be answered based on the Maslov index computation.

Julia Nalven
University of California, Berkeley
Talk title: Modeling shock demagnetization to detect unexploded ordnances

Bombs or shells that fail to explode upon impact with the ground can become buried and pose a threat of detonation even years after being dropped. There is evidence that these unexploded ordnances, which gain a slight magnetization during their construction, undergo demagnetization upon impact with the ground, and modeling the resulting remnant magnetization can be useful in determining whether a given region contains explosive objects rather than harmless pieces of metal. I will outline the analysis of these objects’ energy and atomic structure used to model shock demagnetization.

Maryam Namazi
University of Victoria
Talk title: Effect of barotropic shear on Kelvin waves associated with the first baroclinic mode

Abstract:
The equatorial atmosphere harbours a large spectrum of waves that are trapped near and travel along the equator. These equatorially trapped waves interact nonlinearly with each other and with the planetary-barotropic waves. Kelvin waves, which are observed to play a central role in organized tropical convective systems, are the simplest examples of these equatorially trapped waves. Unlike the uncoupled dry case, Kelvin waves in nature do have a weak but non-zero meridional velocity (Wheeler and Kiladis 1999). A plausible explanation for this comes from the effect of barotropic shear on the Kelvin waves.

Here, we discuss the effect of the two proposed barotropic shear, equatorial westerly shear and equatorial easterly shear on the equatorially trapped Kelvin waves by using three different numerical methods. We compare the performance of the f-wave method, the central scheme and a relaxation type scheme that permits to preserve exactly the meridional balance of the Kelvin wave and its zero meridional velocity, for the non-forced case. We demonstrate that the shear actually induces a weak but non-trivial meridional velocity.

Joel Phillips
McGill University
Talk title: Finite elements on pyramids

Abstract:
I will give a quick overview of Nedelec's high order tetrahedral and hexahedral finite elements for problems that lie in H¹, H(curl), H(div) and L² spaces and then present our construction of high order pyramidal finite elements that are analogous to, and compatible with, them.

Bryan Quaife
Simon Fraser University
Talk title: Integral equation methods for two elliptic operators

Abstract:
I will discuss how integral equation can be used to solve two elliptic operators that arise from temporal discretizations of different partial differential equations. Numerical examples and future work will be discussed.

Marc Ryser
McGill University/Simon Fraser University
Talk title: The cellular dynamics of bone remodeling: a mathematical model

Abstract:
The mechanical properties of vertebrate bone are largely determined by the outcome of the process of bone remodeling, which occurs asynchronously at multiple sites in the mature skeleton. At each location, bone-resorbing osteoclasts and bone-forming osteoblasts are organized in Bone Multicellular Units (BMUs), which contain 10-20 osteoclasts in the leading front followed by 1000-2000 osteoblasts. A BMU exists much longer than individual osteoclasts and osteoblasts, traveling at a rate of 20-40 µm/day for 6-12 months, while maintaining an approximately cylindrical structure of 0.2 x 2 mm. Both osteoclasts and osteoblasts produce multiple paracrine as well as autocrine factors that allow them to interact with each other. In this communication process, the RANKL/OPG pathway has been shown to play an important role.

To analyze BMU progression, we developed a mathematical model that describes the evolution of the osteoclast and osteoblast densities as well as the dynamics of the chemicals RANKL and OPG. The resulting equations, a system of non-linear Partial Differential Equations (PDEs) with appropriate initial and boundary conditions, were studied using a finite difference scheme in space and a 4th order Runge-Kutta scheme in time. We have found that our model successfully recapitulates the spatial and temporal dynamics observed in vivo in a cross-section of the bone. So far, several in silico experiments have been performed and the results allowed us to draw some interesting conclusions about the role of RANKL and OPG, e.g. a possible mechanistic explanation for the absence of backwards branching in BMUs. In the future, this model will allow in silico analysis of the impact of cytokines, growth factors and biomaterials on the process of bone remodeling.

Reinel Sospedra-Alfonso
University of Victoria
Talk title: Global classical solutions to the relativistic Vlasov-Maxwell system with bounded spatial density

Abstract:
The relativistic Vlasov-Maxwell system (RVM in short) describes the time evolution of a collisionless plasma whose particles interact through the electromagnetic field inherent to their charges and their velocities are comparable to the speed of light. The global in time classical solvability for this system remains an open problem. In their ground-breaking result, Glassey and Strauss showed that if the growth in the momentum of the particles is controlled, the RVM system has classical solution globally in time. Subsequently, they proved that such control is achieved if the kinetic energy density of the particles remains bounded for all time. In this talk, we show that the latter assumption can be weakened to the boundedness of the spatial density.

Olga Trichtchenko
McGill University/Simon Fraser University
Talk title: Creating harmonies using neural networks

Abstract:
The goal of this project was to try to teach a neural network to produce a harmony from a given melody. The reason for using a neural network is that making music is a learned process, very much like like a language. However with music, even the accidental occurences can be considered as a form of expression. Using different types of neural networks such as one layer, one layer with bias and two layer, as well as different learning algorithms, the neural networks created for this project were taught to output consonant harmonies. The input for the networks was a series of notes written in binary representation and a harmony was defined to be either a third, fifth or an octave higher than the main voice. After training and running the different algorithms, it was shown that they could all learn to reproduce the melody-harmony pairs they trained on as well as generalize to melodies that were not seen by the networks. However, if the network had more layers, more training data was needed to learn the harmonies. Undertraining also led to interesting harmonies. Using Matlab and midi files, all the melodies and harmonies can be played and analyzed by the listener.

Xiaosheng Zhuang
University of Alberta
Talk title: Matrix extension and its application to wavelets

Abstract:
In this talk, I will talk about matrix extension problem which is closely related to the construction of wavelets. The matrix extension is about how to extend an n-by-r matrix P(z) with Laurent polynomial entries to a whole n-by-n unitary matrix A(z), i.e., A(z)^*A(z)=I. A step by step algorithm will be provided.