Mathematics - Student Works

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Now showing 1 - 5 of 16
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    Operator algebras and symbolic dynamical systems associated to symbolic substitutions
    (2022) Gawlak, Dylan; Ramsey, Christopher
    In 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.
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    Robustness of maximin space-filling designs
    (2021) Lisitza, Cassandra
    In 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.
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    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.
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    Hashtag politics: a Twitter sentiment analysis of the 2015 Canadian federal election
    (2020) Mullins, Amanda; Epp, Adam
    We 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.
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    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.