## Undergraduate Research Assistantships (URAs)

The Undergraduate Research Assistantship (URA) program is a competitive program that provides summer research opportunities to undergraduate students with demonstrated financial need.

Job postings are typically made available in mid-January.

For information on postings and how to apply, visit the URA information page [1] from Student Financial Services.

## NSERC's Undergraduate Student Research Awards (USRAs)

Undergraduate Student Research Awards (USRAs) are meant to stimulate students' interest in research in the natural sciences and engineering. They are also meant to encourage a student to undertake graduate studies and pursue a research career in these fields. If students would like to gain research experience in an academic setting, these awards can provide them with financial support through their host university.

Job postings are typically made available in mid-January.

For information on postings and how to apply, visit the USRA information page [2] from Student Financial Services.

## Advanced Research Project (MATH/STAT*4600)

Each semester, undergraduate students have the opportunity to complete mathematics and statistics research projects with faculty members. This is a terrific opportunity to work on an advanced project that matches your interests and goals. Speak to your instructors about project opportunities and find a research supervisor. You may then register for either MATH*4600 or STAT*4600. These are both 1.00 credit courses that are offered in both the Fall and Winter semesters.

### MATH*4600 Projects

**Extracting E.coli growth parameters from impedance measurements**

(Valerie Hodgins, Fall 2016, Supervisor: Hermann Eberl)

Impedance microbiology is an experimental technique that is based on the observation that certain electrophysical properties of a medium in which bacteria grow are altered by microbial activity. Aim of this project was to develop a mathematical model that allows to determine bacterial growth curve parameters from impedance measurements, and to use it to compare the growth rates of pathogenic and nonpathogenic E.coli strains. This involved methods from ordinary differential equations, biomathematics, optimization, statistics, and programming in R.**Detecting entanglement in Hankel and Toeplitz density matrices**

(Matthew Kazakov, Fall 2016, Supervisor: Rajesh Pereira)

Quantum entanglement is an inherent trait to the universe that has, and continues to, elude mathematicians and physicists. Originally appearing in the equations brought forth by Einstein, and several other quantum pioneers, entanglement is the phenomena where linking two (or more) particles appears to result in an instantaneous exchange of information. This is something that goes against the logic of there being a finite upper bound on the rate at which communication can occur, which is what Einstein and physicists alike got tangled up in. Measuring or altering one part of the entangled system seems to instantly affect other parts of the system.

Mathematically, entanglement can be modelled in the following way; suppose that a Hilbert space is composed of two subspaces, label them and such that . A pure state is unentangled (or equivalently; separable, factorable, etc.) if for and . If cannot be written as a single tensor product, then the state contains some measure of entanglement. Similarly regarding mixed states, a density matrix is separable if it can be written as a summation of tensor products, i.e. for some matrices and within the subsystems of.

This project focused on the analysis of Hankel and Toeplitz density matrices and it was shown that they are positive partial transpose (PPT) and in the case of Hankel matrices, they fulfill the definition of what it means to be separable.

**Mathematical modeling of biofilm growth in porous media**

(Harry Gaebler, Winter 2016, Supervisor: Hermann Eberl)

In porous media, such as soils, bacteria colonize the void space and form so-called biofilms. This principle is used in some environmental engineering techniques, for example in soil remediation, groundwater protection, or wastewater treatment. The aim of this project was to formulate and study a mathematical model that allows to couple microscopic bacterial behavior (growth, detachment, attachment) to macroscopic processes such as flow and pollution transport through the porous medium. This involved methods from ordinary and partial differential equations, advanced calculus, biomathematics, numerical methods, and MATLAB programming.**Exploring Fractals with Contraction Mappings**

(Mark Hirst, Fall 2015, Supervisor: Matt Demers)

A fractal can be defined as a set that is made up of a union of shrunken and distorted copies of itself. A number of different strategies can be applied to generate fractals. The particular methodology used to explore fractals in this project involved the application of contraction mappings on a complete metric space. Banach's fixed point theorem guarantees a unique fixed point in this setting that may be approximated through direct iteration upon any seed point in the space. When the complete metric space is a space of sets, and the contraction mappings are taken to be an iterated function system (which we will call an IFS -- that is, a union of set-valued contraction mappings), the unique fixed point that is obtained through iteration upon a seed set is called the attractor, typically a fractal set. We can choose different mappings to generate different fractal sets in a variety of settings; for example, we generated fractal sets in two and three dimensions.

There is a continuity result associated with IFS: It can be shown that if two IFS are "near" to one another, their attractors will be somewhat similar. With this in mind, we found that by gradually changing the parameters defining one IFS to those defining a second IFS (thereby, with each incremental change, obtaining a new IFS which could be iterated), we obtained fractal sets that gradually changed in appearance from the first attractor to the second attractor. In this way, we are able to speak of families of fractals that exist as fixed points of a series of IFS that in some sense lie "between" any two existing IFS.**The Choquet Integral**

(Coralie Escot, Fall 2015, Supervisor: Rajesh Pereira)

Coralie examined an approach to the Lebesgue integral first proposed by Gustave Choquet and showed how it can be used to prove classical theorems such as Fatou's lemma and the Monotone Convergence Theorem in a fairly easy way, essentially making this formulation accessible to undergraduate students.**Modifying the Traffic Model: Creating an Algorithm for Driver Response to Creatures Crossing the Highway**

(Brady Dortmans, Winter 2015, Supervisor: Anna Lawniczak)

We investigate the model of cognitive agents simulated to act as naïve creatures as they attempt to cross the highway and avoid being killed by cars. These creatures learn by observation and are unable to determine vehicle motion analytically but are able to approximate the vehicle motion. The creatures are born as “tabula rasa” i.e. no previous knowledge of the environment and build their own knowledge base. The creatures have a formula mechanism using what they observe, along with traits of fear and desire, to determine if it is safe to cross the highway. The model’s algorithm is changed to allow for vehicles to respond to a creature’s presence on the highway and an extra waiting period to allow creatures to study the change in car motion. The cars are governed by a modified Nagal Schreckenberg model and a special case model when a creature is seen.**Evolving Self-Driving Automata to Distinguish Classes of DNA**

(Sierra Gillis, Fall 2015, Supervisor: Dan Ashlock)

The motivation for this project came from the abundance of bacteria that cannot be cultured but whose genomes can be sequenced. When sequencing bacterial data from a soil or water sample with unknown amounts and types of bacteria, programs are needed to classify or bin the DNA according to its features. This can give an estimate of the number of groups of bacteria present. My supervisor, Daniel Ashlock, proposed using evolution to optimize Self-Driving Automata (SDA) for performing this classification. My research in particular was to develop a way of finding features in the DNA and to investigate the parameters involved in the evolutionary algorithm, for example mutation rate and population size. The SDAs are capable of producing an infinite string of bases, which is then compared base-wise to the DNA. The fitness of an SDA is based on the amount of matches and mismatches between its infinite string and the target DNA sequence. We discovered that the fitness function we used lead to the SDAs being no better at distinguishing classes than pre-existing methods. While this was a setback, it gave me a great perspective of typical research.

### STAT*4600 Projects

**Monitoring fork lengths of Athabasca white sucker: A simulation study**

(Neil Faught, Fall 2016, Supervisor: Lorna Deeth)

Researchers are currently sampling various species of fish from the Athabasca River in northern Alberta to determine what effects, if any, oil sands development has had on their populations. As of now, researchers have analyzed yearly physiological measurements of sampled fish to see if any change has occurred their physiology that may be a result of human activity. As there are many statistical monitoring methods available for performing such a task, this report applied a handful of such methods (such as moving average and exponentially weighted moving average models) to simulated data to determine what model-parameter set combination was most successful at detecting changes in this simulated data. Scoring metrics had to be developed to evaluate the performance of each model-parameter set combination tested.**Estimating Abundance of Seabirds**

(Denys Kelly, Winter 2014, Supervisor: Julie Horrocks)

Abundance estimation of species is becoming increasingly important with the world’s rapidly changing environment. This project investigated the effects of various environmental and geographic variables on the abundance of two species of seabirds, the Northern Gannet (NOGA) and the Greater Shearwater (GRSH), from 2008 to 2013. A seabird observation dataset provided by Environment Canada’s Canadian Wildlife Service program, Eastern Canada Seabirds at Sea (ECSAS) was restructured and merged with environmental data available from Oregon State University. A variety of statistical models were fit to these data and the following environmental and geographical variables were found to be important: latitude, longitude, distance from shore, bathymetry, primary production and the time of year. Models were then used predict abundance of seabirds at specific latitude and longitudes.

**A dissimilarity measure for comparing MaxEnt models**

(Jeffrey Daniel, Winter 2013, Supervisor: Gary Umphrey)

Mapping the geographic distribution of plants and animals is a topic of major interest in ecology. A species distribution model is a statistical model that relates species presence/absence data to environmental data in order to estimate the environmental conditions that define a species’ habitat, and plot that habitat on a map. One of the most popular species distribution modelling tools is the software package MaxEnt. In this project, I developed a method for comparing the output of two MaxEnt models to identify regions where the models disagree and to quantify the amount of disagreement. I then used this method to assess how MaxEnt models are affected by sample size and sampling bias.