# Mathematics - Student Works

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Item Characterizing semi-Dirichlet algebras and their graphs(2024) Rumball, JustinIn this paper, we relate certain finite-dimensional operator algebras to graphs and define their equivalent properties. Using these graph's properties, we will determine which algebras are semi-Dirichlet and how many non-unitarily similar semi-Dirichlet algebras there are corresponding to graphs with up to five vertices.Item Use of LaTeX and other software for illustrating mathematical concepts(2024) Jagal, Tyler; Solomonovich, MarkThe work is focused on the use of Latex and related graphical software for illustrating some mathematical concepts. Those concepts include graphical presentations related to some notions of analysis in the extended complex plane. In particular, special considerations will be given to the stereographic projection, which often provides a natural setting for interpreting transformations in the complex plane.Item Generating functions related to the Fibonacci substitution(2023) Pouti, Aisling; Phan, NhiIn this paper, two generating function representations of the Fibonacci Substitution Tiling are derived and proven to converge on the interval -1Item Operator algebras and symbolic dynamical systems associated to symbolic substitutions(2022) Gawlak, Dylan; Ramsey, ChristopherIn this thesis, we shall look at symbolic dynamical systems and operator algebras that are associated with these systems. We shall focus on minimal shift dynamical systems generated by symbolic substitutions. By first characterizing the shift spaces associated to primitive substitutions, we shall see that all minimal dynamical systems generated by symbolic substitutions are conjugate to proper primitive substitutions. In doing this, we will look at ordered Bratteli diagrams and strongly maximal TAF-algebras and we will see how we can associate these to a type of dynamical system called a Cantor minimal system, of which infinite minimal shift spaces are an example. We also shall see how we can associate a semi-crossed product algebra to a topological dynamical system and how isomorphism of two semi-crossed product algebras is equivalent to conjugacy of their associated dynamical systems. The semi-crossed product algebra is more general in that it can be associated to any dynamical system, whereas TAF-algebras can only be associated to Cantor minimal systems.Item Robustness of maximin space-filling designs(2021) Lisitza, CassandraIn this report, we first have a review of the maximin space-filling design methods that is often applied and discussed in the literature (for example, Müller (2007)). Then we will discuss the robustness of the maximin space-filling design against model misspecification via numerical simulation. For this purpose, we will generate spatial data sets on a n x n grid and design points are selected from the n2 locations. The predictions at the unsampled locations are made based on the observations at these design points. Then the mean of the squared prediction errors are estimated as a measure of the robustness of the designs against possible model misspecification. Surprisingly, according to the simulation results, we find that the maximin space-filling designs may be robust against possible model misspecification in the sense that the mean of the squared prediction error does not increase significantly when the model is misspecified. Although the results were obtained based on simple models, this result is very inspiring. It will guide further numerical and theoretical studies which will be done as future work.Item Lie groups and quantum chromodynamics(2020) Davey, Aaron C. H.This article explores the mathematics of group theory, in particular Lie groups and their representations, and its connection to quantum field theory, specifically quantum chromodynamics. The first half discusses specific Lie groups and their Lie algebras and uses this information to describe the theory of the strong force, its particle constituents and a property of subatomic particles known as colour charge. This article emphasizes the importance of utilizing abstract algebra for the continued advancement of quantum physics in the modern era.Item Hashtag politics: a Twitter sentiment analysis of the 2015 Canadian federal election(2020) Mullins, Amanda; Epp, AdamWe developed a split plot design model for analysis of sentiment toward federal political parties on the social media platform Twitter in the weeks prior to the 2015 Canadian Federal Election. Data was collected from Twitter’s Application Programming Interface (API) via statistical program R. We scored the sentiment of each Twitter message referring to the parties and tested using ANOVA. Our results suggested that the Liberal Party and New Democratic Party had more positive sentiment than the Conservative Party. Actual seat wins coincide with our results for the Liberal Party (which won 148 new seats) and the Conservative Party (which lost 60 seats), but positive sentiment for the New Democratic Party did not correspond to seat wins.Item Schrodinger wave equation(2020) Davey, Aaron C. H.The father of quantum mechanics, Erwin Schrodinger, was one of the most important figures in the development of quantum theory. He is perhaps best known for his contribution of the wave equation, which would later result in his winning of the Nobel Prize for Physics in 1933. The Schrodinger wave equation describes the quantum mechanical behaviour of particles and explores how the Schrodinger wave functions of a system change over time. This project is concerned about exploring the one-dimensional case of the Schrodinger wave equation in a harmonic oscillator system. We will give the solutions, called eigenfunctions, of the equation that satisfy certain conditions. Furthermore, we will show that this happens only for particular values called eigenvalues.Item Diffraction of fully Euclidean model sets(2021) Klick, Anna; Strungaru, NicolaeWe provide a elementary proof, using only the Poisson summation formula and theory of tempered distributions, of the well-known fact that fully Euclidean regular model sets produce a pure point diffraction measure.Item The two physics governing the one-dimensional cubic nonlinear Schrödinger equation(2021) Mucalica, Ana; Bica, IonIn 1926, in his quest to explain the quantum probabilistic nature of particles, Erwin Schrödinger proposed a nonrelativistic wave equation that required only one initial condition, i.e., the initial displacement of an electron. His equation describes the wave-particle duality discovered by Louis de Broglie in 1924. Furthermore, Schrödinger's wave equation is dimensionless, allowing the equation to be a mathematical model describing different physical phenomena. Introducing nonlinearity into the Schrödinger equation, we worked with the so-called self-focusing nonlinear Schrödinger equation. We showed that when the nonlinearity is perfectly balanced with the dispersion, the self-focusing nonlinear Schrödinger model describes the propagation of a soliton. In 1968 Peter Lax introduced the "Lax Pair," a pair of time-dependent matrices/operators describing the nature of a nonlinear evolution partial differential equation, to discuss solitons in continuous media. This procedure is what we call the scattering method for describing mathematically nonlinear processes in physics. We used the scattering method to find the Lax Pair for the nonlinear Schrodinger model, and we showed that the equation is a compatibility condition for the AKNS system. In 1974, Ablowitz, Kaup, Newell, and Segur (AKNS) introduced the inverse scattering transform to solve evolution nonlinear partial differential equations arising from compatibility conditions for the AKNS system. Rather than using the inverse scattering transform, we showed an intuitive approach in revealing the formation and propagation of a soliton for the self-focusing nonlinear Schrodinger equation, using a novel approach via cnoidal waves. The work will also include a novel theorem describing the steepening of the wavefront due to nonlinearity.Item Lorenz’s system - analysis of a sensitive system(2021) Lisitza, Cassandra; Bica, IonMeteorology is a branch of geophysics concerned with atmospheric processes and phenomena and atmospheric effects on our weather. Edward Lorenz was a devoted meteorologist who made several significant contributions to this field. We first describe the fascinating history of Lorenz’s discoveries and his revolutionary additions to the area of meteorology. In particular, he noted the extremely sensitive dependence on the initial conditions of a Chaotic system in the atmosphere, which is commonly referred to as the Butterfly Effect and pertains to Lorenz’s system of three Ordinary Differential Equations (ODEs) that models the atmospheric convection in the atmosphere. We conducted a novel, in-depth mathematical analysis of the Theorem of Existence and Uniqueness for a system of ODEs in general and addressed how it applies to Lorenz’s system. Further, we exhibited how Lorenz’s system is ill-posed using an application where we varied the initial parameters by minimal variations and noted relatively quick and drastic differences in the trajectories of the system.Item Van der Pol oscillator – analysis of a non-conservative system(2020) Reeves, Adam; Bica, IonThe Van der Pol oscillator was introduced by Balthasar van der Pol, who was ”a famous scholar, a famous scientist, a famous administrator at the international level, he was equally well known for the clarity of his lectures (in several languages), his knowledge of the classics, his warm personality and his talents for friendship, and his love for music.” [2] The oscillator describes the nonlinear oscillations for systems like a triode circuit, which produce selfsustained oscillations known as relaxation oscillations. Extensive studies have been done on the oscillator, for understanding it and for using it as an applied model for the heartbeat, for example. In this thesis, we will explain the nature of the oscillator from an original point of view, in the low-friction regime. First, we will give an intuitive physical explanation of the first order averaging method, a perturbation theory method, applied onto the oscillator. We will follow with an analytical approach of the first order averaging method, and we will show the mathematical complexity of it. We will conclude with the application of the first order averaging method to the Van der Pol oscillator, confirming the findings from the intuitive approach.Item Van der Pol oscillator – analysis of a non-conservative system(2020) Reeves, Adam; Bica, IonThe Van der Pol oscillator was introduced by Balthasar van der Pol, who was ”a famous scholar, a famous scientist, a famous administrator at the international level, he was equally well known for the clarity of his lectures (in several languages), his knowledge of the classics, his warm personality and his talents for friendship, and his love for music.” The oscillator describes the nonlinear oscillations for systems like a triode circuit, which produce self-sustained oscillations known as relaxation oscillations. Extensive studies have been done on the oscillator, for understanding it and for using it as an applied model for the heartbeat, for example. In this thesis, we will explain the nature of the oscillator from an original point of view, in the low-friction regime. First, we will give an intuitive physical explanation of the first order averaging method, a perturbation theory method, applied onto the oscillator. We will follow with an analytical approach of the first order averaging method, and we will show the mathematical complexity of it. We will conclude with the application of the first order averaging method to the Van der Pol oscillator,confirming the findings from the intuitive approach.Item Hydrostatic balance in meteorology(2020) Mucalica, Ana; Bica, IonMeteorology is a branch of geophysics that studies the properties of the atmosphere that “cushions” the Earth and all the phenomena that happen within it. The hydrostatic balance occurs when the pressure at any point in the fluid equals the weight of an air column of the unit section from above the point, and in these ideal conditions, we have the hydrostatic equation for the fluid in hydrostatic balance. The hydrostatic equation is viewed as the hydrostatic equilibrium condition, which provides an accurate approximation for the vertical dependence of the pressure field in the real atmosphere. In this talk, we manipulate a mathematical model describing the variation of pressure to altitude in the atmosphere using partial differential equations, and we will show how realistic charts based on real data are obtained, and compare them to charts obtained by using a climate change scenario when the temperature at sea-level is warmer by 2℃ than the one used in current charts. Work done in collaboration with Anneliese Ansorger, Cory Efird, Cassandra Lisitza, Ghristopher Macyk, Tarig Mergani, Adam Reeves, Rebecca Walton, Yaying Zhong, and Jett Ziehe, MacEwan University.Item On arithmetic progressions in model sets(2022) Klick, Anna; Strungaru, Nicolae; Tcaciuc, AdiWe establish the existence of arbitrary-length arithmetic progressions in model sets and Meyer sets in Euclidean d-space. We prove a van der Waerden-type theorem for Meyer sets. We show that subsets of Meyer sets with positive density and pure point diffraction contain arithmetic progressions of arbitrary length.Item Where do students rest at MacEwan University: A mini-research project by Terence Kamau(2018) Kamau, Terence; Buro, KarenIn university and colleges, napping is a common occurrence and practice used by students to alleviate daytime sleepiness. Sleep debt up to 19 hours is on par with a drunk individual having a blood alcohol content (BAC) of 0.05 (two wine glasses), whereas most states have the BAC limit at 0.08 for non-commercial and 0.04 for commercial drivers. Napping stops the symptoms, where there’s too little time to cure the restlessness. MacEwan University will be observed for proportions of students napping on furniture and the proportion of total seats taken in the same area. Data was collected twice a day (once Monday) for 7 days. Areas were observed for an hour interval, at four different times. Furniture seating was counted through furniture design. Five areas were observed for 10-minute periods for napping individuals and total seats occupied. Sex of the individual presumed male, presumed female, ambiguous was observed. Sound level of each area was recorded by a sound app for a one-minute interval. Area description will account for furniture differences for suitability of napping area. A binary logistic regression was used to analyze the data, which showed location and time as significant to where people sat (p-value<0.05).Item Where do students rest in MacEwan University(2018) Kamau, Terence; Buro, KarenIn University/College institutions, napping is a common occurrence and practice used by students to alleviate daytime sleepiness. Sleep debt up to 17 to 19 hours is on par with a drunk individual having a blood alcohol content (BAC) of 0.05 (two wine glasses), whereas most states have the BAC limit at 0.08; 0.04 for commercial drivers. Napping stops the symptoms, where there’s too little time to cure the disease. MacEwan University will be observed for nap proportions of students napping on furniture, to the proportion of total seats taken in the same area. Data was collected twice a day (once Monday) on the weekday, for 7 days. Areas were observed for an hour interval, at four different times. Furniture seating was counted through furniture design. Five areas were observed for 10-minute periods for napping individuals and total seats occupied. Sex of the individual presumed male, presumed female, ambiguous was observed. Sound level of each area was recorded by a sound app for a one-minute interval. Area description will account for furniture differences for suitability of napping area. A binary logistic regression was used to analyze the data, which showed location and time as significant to where people sat (p-value<0.05).Item Schrodinger wave equation(2018) Davey, Aaron C. H.; Zadorozhna, NataliyaThe father of quantum mechanics, Erwin Schrodinger, was one of the most important figures in the development of quantum theory. He is perhaps best known for his contribution of the wave equation, which would later result in his winning of the Nobel Prize for Physics in 1933. The Schrodinger wave equation describes the quantum mechanical behaviour of particles and explores how the Schrodinger wave functions of a system change over time. This project is concerned about exploring the one-dimensional case of the Schrodinger wave equation in a harmonic oscillator system. We will give the solutions, called eigenfunctions, of the equation that satisfy certain conditions. Furthermore, we will show that this happens only for particular values called eigenvalues.Item MacEwan University WiFi analysis(2016) Prince, James; Yong, Alan; Anton, CristinaOne of the worst feelings in the world is waiting for a slow Internet connection. While this may be more a reflection of our impaired society than a faulty modem, this study will shed some light as to soothe these pains while on MacEwan University Campus. MacEwan University has recently undergone a "WiFi Renovation", with many new WiFi units installed all throughout the school. The goal of this experiment is to find where in the school are the strongest and weakest connections. This will be an interesting reflection on the new system effectiveness and coverage. The factors that will be tested for are the Location in the school, Time of Day, Day of the Week and Type of Device used, and blocking will be done on the last four factors. To measure the connection quality, a file of a pre-determined size will be downloaded and the time taken will be recorded. Alan will be using an Apple Iphone, and James will be using a Samsung Galaxy S3, which will eliminate the chance of a newer device being different than an older one, or an Apple device vs an Android. This study will determine which factors are significant, and which factor combination yields the best results in terms of WiFi connectivity. This method of mapping a WiFi system will be useful to students and to the IT management of the University because the results of this study will provide the school with information which will help plan for future changes to WiFi layout. An easy extension of the methodology of this experiment could be developed and used to assess any WiFi or cellular device service. The results from this experiment alone will be interesting, but a larger application of the method could be groundbreaking.