## Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS

Microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS), which combine electronics with miniature-size mechanical devices, are essential components of the modern technology. In order to provide accurate, controlled, and stable locomotion for such microdevices, researchers have proposed a variety of models, based upon thermal, biological, or electrostatic forces. There are many industries and manufacturers in Ontario who rely on MEMS technology in a crucial way. Our objective is to use mathematical analysis to predict various phenomena related to some of these models.

## Mathematical modeling of Listeria overgrowth phenomenon

The standard method to test for Listeria n7onocj'logenes in foods is prone to false negatives if also non-pathogenic members of the Listeria genus are present. These can overgrow L.monocylogenes during the selective enrichment process and mask the presence of the pathogen. This "overgrowth'" phenomenon is not well understood, partially because the mathematical models that are used in food microbiology to determine growth kinetics of bacteria are not able to capture interaction between species correctly.

## Pediction of the Occurrence of Solar Eruptions

The Earth's environment is affected by space weather conditions. The most profound effects on the Earth are driven by solar eruptions, such as solar flares and solar storms. Such phenomena produce magnetic disturbances in the upper layers of the Earth's atmosphere, which can be powerful enough to adversely affect electronic equipment. For example, the instrumentation on orbiting satellites can be irreversibly damaged.

## Wavelet Methods for Optimal Control Problems in Pharmaceutical Research

Optimal control problems and reaction-diffusion systems have many applications in pharmaceutical research. For instance, the enzymatics collagen matrix degradation and the principal process governing drug transport inside a solid tumor could both be described by reaction-diffusion systems. Solving these systems using effective numerical algorithms will benefit drug product design, drug delivery and cancer treatment. Since the traditional numerical solutions to these systems require large memory and long computational times, we will consider wavelet methods which allow us to combine high

## Performance Estimation of Davis Hydro Turbine Using 3D Numerical Simulations

The goal of the proposed project is to develop a computational fluid dynamics model of the Davis hydro turbine, so that its performance can be assessed and improved before it becomes a commerically available product. A numerical implementation of the model will be used to estimate important characteristics of the turbine, such as its power coefficient, power quality, and average torque. The reliability of the mathematical model will be tested by comparing the numerical results to the experimental ones.

## Optimization of Road Alignment Design

In the road construction process, a civil engineer commonly uses software to outline the horizontal and vertical road alignment on a topographical map. The software then calculates the amount of earth that needs to be excavated, or filled, at certain points of the alignment as well as pavement costs, land costs and other expenses. Softree, the partner company, provides such software. There is currently no commercial software that offers the user an automated optimization of the alignment based on the total cost.

## Geo-spatial Data Mining Applications in Marketing

The project will develop methods for constructing profiles and predictive models for geo-spatial data. The socio-demographic profile will be developed for the geographical areas of interest. In addition, the models will be developed to identify areas with high potential; for acquiring new customers. The targeting of these areas may be conducted through unaddressed direct mail. Methods of testing effectiveness of marketing campaign and predictive models will be developed.

## Applications of Covering Arrays in Cyber Risk and Security Compliance

With the integration of once independent control systems into business networks there lies a major security risk that organizations may not be prepared to assess or manage effectively. Without proper testing, these real-time systems are vulnerable to attack which creates a significant risk to the reliability and integrity of these systems. With the use of a mathematical object, the Error Locating Array (ELA), we are able to detect errors in the system whenever the structure of the faulty interactions satisfies certain reasonable assumptions.

## Parallel Multivariate Polynomial Multiplication and Division

In this internship, the team proposes to develop high performance sequential and parallel algorithms for multiplication and division of multivariate polynomials. They propose to use a recursive data structure. One of the possible advantages of a recursive data structure is that we can also see how to parallelize polynomial division. Potentially high level algorithms such as computing polynomial GCDs and polynomial factorization will benefit from this speedup.

## Computation of Risk Measures via Efficient Least-squares Monte Carlo

The computation of risk profiles for financial products and portfolios is an extremely important problem, both for regulatory and internal management purposes. For complex products whose value depends on a number of underlying risk factors and for which exercise decisions can be made prior to maturity, Monte Carlo simulation techniques are the only viable procedures. This project aims to adopt a simulation method used for pricing products, to computing risk exposures. Various ways of improving the computational speed will be explored.