The circulation in the Black Sea characterized by a strong basin-wide current along the shore is subject to mesoscale variability in the form of meanders, intense jets, eddies and filaments. We present the results of a new series of laboratory experiments where the typical features of the circulation were modeled. The dynamical similarity of the important dimensionless parameters governing the dynamics of the system was satisfied in our experiments. The results demonstrate the development of the baroclinic instability due to the fresh water discharge imitating the river inflow in the Black Sea. Persistent transient features of the circulation, such as the so-called Batumi Eddy and the Sevastopol Eddy as well as other features are reproduced in our experiments when the background rotation rate of the system is varied. Quantitative data were obtained for the observed velocity and vorticity fields. The results of the experiments are in agreement with observational data as well as numerical modeling.
Thrombin is best known for its role in the chemical reactions leading to the formation of fibrin, one of the structural components of a blood clot. The production of thrombin involves two main pathways, one of which requires positive feedback by thrombin. A mathematical model of thrombin generation in the absence of this feedback pathway was developed to investigate the production of thrombin in human ovarian follicular fluid, an avascular system where the presence of most of the coagulation proteins has recently been discovered. This model consists of 54 nonlinear ordinary differential equations describing the concentrations of the various proteins and lipid involved. Simulations of the model equations help to determine the controlling factors in restricting thrombin levels in follicular fluid, as well as suggest various laboratory experiments that should be conducted to elucidate the role of thrombin in this system. Applications of this work include determining a probable role for thrombin in folliculogenesis and the implications of novel anticoagulant therapies on fertility.
The dynamic nature of nuclear architecture within eukaryotic cells is an interesting problem in cellular biology. Specifically, the origin, maintenance and disappearance of speckles, which are heterogeneously distributed nuclear compartments enriched with pre-mRNA splicing factors, is unknown. It has been hypothesized that a process of self-aggregation among dephosphorylated splicing factors, modulated by a phosphorylation-dephosphorylation cycle, is responsible for the formation and disappearance of speckles. Also, it is thought that the existence of an underlying nuclear structure plays a major role in the organization of splicing factors.
We will explain how these hypotheses and a diffusion-approximation approach allow for the derivation of a fourth order aggregation-diffusion model that describes a possible mechanism underlying the organization of splicing factors in speckles. A linear stability analysis, supplemented with numerical simulations, will show how the self-interaction among dephosphorylated splicing factors can result in spatial patterns that are caused by instabilities about homogeneous steady states.
Directed cell movement in response to an external chemical gradient is involved in many diverse physiological processes such as wound healing and embryogenesis. The small G proteins, a class of proteins known to regulate actin polymerization, establish intracellular concentration gradients in response to external stimuli. Enhanced polymerization of the protein actin at the leading edge of the cell generates sufficient force for membrane protrusion and movement. Using mathematical modelling, we simulate the orientation and density of actin filaments and polymerizing filament tips against a rigid obstacle. We compare our results to experimental data and discuss various hypotheses regarding the formation of the actin network and regulation of actin polymerization at the leading edge of a motile cell.
We developed a new reconstruction method called the inverse polynomial reconstruction method for the resolution of the Gibbs phenomena. It is well known that the Fourier approximation of nonperiodic or discontinuous function or the polynomial approximation of discontinuous function in a given interval suffers from the so-called Gibbs phenomena. The convergence of the Fourier approximation in the smooth region is O(1/N) and O(1) in the neighborhood of discontinuity in its maximum norm due to the Gibbs oscillations.
The proposed inverse method seeks a polynomial reconstruction such that the residue between the Fourier representations of the reconstructed function and the original function is orthogonal to the Fourier space. Consequently the reconstrucion is uniquely determined and the method yields spectral accuracy removing the Gibbs oscillations completely. Furthermore the reconstruction is exact if the original function is a (piecewise) polynomial. We provide some numerical examples for both one and two dimensional problems.
This is joint work with Bernie D. Shizgal at the University of British Columbia.
We present stability of switching systems with time delay by using Lyapunov Functional and Lyapunov Function. We extend results of ODE switching systems to stabilize a class of DDE switching systems given a proper condition on the time delay.
Far field wake behind a towed self-propelled body is simulated by using a point force/force doublet idealization. These wakes become unstable and form vortex streets. A new series of high-resolution 2D numerical simulations is performed to study the characteristics of the wakes including the shedding frequency for a wide range of control parameters such as translational velocity, magnitude and spatial extent of a localized force.
The results of numerical experiments of unsteady wake flow show the existence of a great variety of flow regimes and are in good qualitative agreement with preliminary laboratory experiments.
The particle boundary layer (PBL), a new terminology, is defined as the region in which the concentration profile of particles has a noticeable variation in depth. Although particle-driven gravity currents are commonly modelled by assuming a vertically well-mixed particle distribution, such PBL exists theoretically, especially for larger size of particles. Similar observations are also obtained from experiments; a highly stratified density profile of a low-concentration flow has been observed, particularly near the surface of solid bodies such as a river bed. To fully understand the dynamics of the flow within the PBL, of interest will be constructing a mathematical model to investigate how the particle distribution and scalings in the governing equations are affected within such a layer. In addition, the typical thickness of the PBL can be approximated algebraically from the transport equation, which is then compared with that of the viscous boundary layer. The ratio of the thickness of the two layers eliminates the viscous terms from the Navier-Stokes equations resulting in a system of first-order partial differential equations. Theoretical solutions of the system will be shown and discussed in several cases using perturbation theory and method of characteristics. These results are then compared with numerical solutions obtained using finite-difference scheme.
We use a simple compartmental model to show a possible mechanism for multiple outbreaks or even sustained periodic oscillations of emerging infectious diseases due to the psychological impact of reported numbers of infective and hos pitalized individuals. This impact leads to the change of avoidance and contact patterns at both individual and community levels, and incorporating this impact using a simple nonlinear incidence function into the model shows qualitative differences of the transmission dynamics.
A probability density function (PDF) fumigation model is presented here to study the dispersion of air pollutants emitted from a tall stack on the shoreline. This work considers dispersion of the pollutants in the stable layer and within thermal internal boundary layer (TIBL) proceeds independently. The growth of TIBL is considered parabolic with distance inland and turbulence is taken as homogeneous and stationary within the TIBL. Dispersion of particles (contaminant) in lateral and vertical directions is assumed independent of each other. This assumption allows us to consider the position of particles in both directions as independent random variables. The lateral dispersion distribution within the TIBL is considered as Gaussian and independent of height. A skewed bi-Gaussian vertical velocity (w) PDF is used to account for the physics of dispersion due to different characteristics of updrafts and downdrafts within TIBL. Incorporating finite Lagrangian time scale for the vertical velocity component, it is observed that it reduces the vertical dispersion in the beginning and moves the point of maximum concentration further downwind. Due to little dispersion in the beginning, there is more plume to be dispersed causing higher concentrations at large distances. The model has considered Weil and Brower's (1984) convective limit to analyze dispersion characteristics within TIBL. The revised model discussed here is evaluated with the data available from the Nanticoke field experiment on fumigation conducted in the summer of 1978 in Ontario, Canada. The results of the revised model are in better agreement with the observed data. The paper suggests the use of mean absolute error and mean relative error as quantitative measures of model performance along with the residual analysis.
We probe the micron-scale rheology of Carbopol ETD2050, a polymer with a wide range of industrial applications, by studying the motion of small suspended tracer particles. With the aid of video fluorescence microscopy and particle tracking software, we record the positions of several tracer particles simultaneously and extract the local viscoelastic properties of the fluid from their dynamics. We study these properties as a function of polymer concentration and tracer particle diameter.
Accurate assessment of the volume of cerebral ventricles on computed tomographic images of the brain is an important and as yet unsolved problem in neuroradiology. Current subjective assessment of ventricles by neuroradiologists and neurosurgeons has limited accuracy, because of the complex shape of the ventricular system. We are developing an automated system that can segment the cerebral ventricles on axial computed tomographic images of the brain. Two automated segmentation techniques have been developed and tested. One is based on thresholding and the other on region growing. The results have been compared to a manual segmentation by calculating the similarity index (S). A good result (S > 0.7) was obtained.
This poster will present new results regarding symmetric binary fractal trees, obtained using methods of computational topology. Fractal trees are interesting mathematical objects and they have many biological applications. A symmetric binary fractal tree, as first introduced by Mandelbrot, has two parameters that define the branching: the scaling ratio r (with 0 < r < 1) and a branching angle q (with 0 < q < 180). We study the structure of the self-avoiding and self-contacting trees with an analysis of the corresponding e-neighbourhoods (the tree along with points that are with e). The homology of the e-neighbourhoods changes as e® 0, and depends on the values of r and q. This provides new classifications of the trees, and new topological critical points. We also compare growth rates of holes with the scaling dimension.
In a variety of physical phenomena, we often want to track the motion of an interface. Such phenomena can occur in fluid mechanics, material science, medical science, control theory, and image processing. In this poster, we introduce, analyze, and utilize level set methods for the study of such problems. Level set methods are powerful numerical techniques for tracking the motion of interfaces moving in complex ways. They are based on computing solutions to approximate the equations of motion of the fluid. They use techniques borrowed from hyperbolic conservation laws.
One third of the power transmission line outages are caused by extreme weather in Alberta. The lightning is one of the major adverse weather conditions that can cause frequent power system outages. However, the characteristics of lightning and the relationship between lighting and power system outages are still not fully understood by power system planning engineers. This research is to reveal the spatial and temporal patterns of lightning in Alberta. It is found that both the spatial and temporal distributions of the lightning are not uniform. The area with the highest lightning frequency and the highest lightning occurrence days are located at central Alberta along Rocky Mountains. And the lightning frequencies and the lightning occurrence days have decreasing trends from this center to the other parts of Alberta. Meanwhile, the geographical and temporal characteristics of lightning-caused transmission line outages on several voltage levels are being studied. The underlying distribution of outage duration hours on each voltage level are significantly different so that any statistical analysis of the outages has to be carried out separately for each of the voltage levels. Furthermore, linear models between the outage duration hours and the lightning weather elements, by using the best subset selection procedure and Jackknife cross validation method, were established.
Computer modeling of complex phenomenon now plays an important role in all areas of science and engineering. It is often the case that such models are based upon complicated systems of differential equations. MIRKDC [Enright, Muir] is a software package for the numerical solution of systems of first order, nonlinear, boundary value ordinary differential equations, with separated boundary conditions. My research involves modification of the MIRKDC software package in order to incorporate a number of significant performance enhancements including analytic derivative assessment, computational derivative approximation, problem sensitivity (conditioning) assessment, defect control improvement and an auxiliary global error indicator.
Contact rate and quarantine rate in the modeling of infectious diseases are very important. Yet it is always not easy to identify the values for these parameters. In this talk, we develop a method, Optimizing the Objective Function Method, to estimate the multi-parameters. By using dynamical model and the limited data available, we can obtain accurate approximation of the parameters.