Mathematics - Student Workshttps://hdl.handle.net/20.500.14078/672023-09-22T19:12:10Z2023-09-22T19:12:10Z171Generating functions related to the Fibonacci substitutionPouti, AislingPhan, Nhihttps://hdl.handle.net/20.500.14078/32262023-09-15T09:00:23Z2023-01-01T00:00:00Zdc.title: Generating functions related to the Fibonacci substitution
dc.contributor.author: Pouti, Aisling; Phan, Nhi
dc.description.abstract: In this paper, two generating function representations of the Fibonacci Substitution Tiling are derived and proven to converge on the interval -1<x<1. A sequence of signs for the Fibonacci Substitution is established along with a conjecture that the interval of convergence has an infinite number of zeroes.
2023-01-01T00:00:00ZOperator algebras and symbolic dynamical systems associated to symbolic substitutionsGawlak, Dylanhttps://hdl.handle.net/20.500.14078/26412023-02-13T16:33:44Z2022-01-01T00:00:00Zdc.title: Operator algebras and symbolic dynamical systems associated to symbolic substitutions
dc.contributor.author: Gawlak, Dylan
dc.description.abstract: 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.
2022-01-01T00:00:00ZRobustness of maximin space-filling designsLisitza, Cassandrahttps://hdl.handle.net/20.500.14078/26122023-02-11T01:48:48Z2021-01-01T00:00:00Zdc.title: Robustness of maximin space-filling designs
dc.contributor.author: Lisitza, Cassandra
dc.description.abstract: 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.
2021-01-01T00:00:00ZLie groups and quantum chromodynamicsDavey, Aaron C. H.https://hdl.handle.net/20.500.14078/25892023-02-11T01:48:02Z2020-01-01T00:00:00Zdc.title: Lie groups and quantum chromodynamics
dc.contributor.author: Davey, Aaron C. H.
dc.description.abstract: 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.
2020-01-01T00:00:00ZHashtag politics: a Twitter sentiment analysis of the 2015 Canadian federal electionMullins, AmandaEpp, Adamhttps://hdl.handle.net/20.500.14078/25852023-02-11T01:49:39Z2020-01-01T00:00:00Zdc.title: Hashtag politics: a Twitter sentiment analysis of the 2015 Canadian federal election
dc.contributor.author: Mullins, Amanda; Epp, Adam
dc.description.abstract: 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.
2020-01-01T00:00:00ZSchrodinger wave equationDavey, Aaron C. H.https://hdl.handle.net/20.500.14078/25882023-02-11T01:48:47Z2020-01-01T00:00:00Zdc.title: Schrodinger wave equation
dc.contributor.author: Davey, Aaron C. H.
dc.description.abstract: 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.
2020-01-01T00:00:00ZDiffraction of fully Euclidean model setsKlick, Annahttps://hdl.handle.net/20.500.14078/24322023-03-27T15:38:50Z2021-01-01T00:00:00Zdc.title: Diffraction of fully Euclidean model sets
dc.contributor.author: Klick, Anna
dc.description.abstract: We 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.
2021-01-01T00:00:00ZThe two physics governing the one-dimensional cubic nonlinear Schrödinger equationMucalica, Anahttps://hdl.handle.net/20.500.14078/24312023-02-11T01:50:23Z2021-01-01T00:00:00Zdc.title: The two physics governing the one-dimensional cubic nonlinear Schrödinger equation
dc.contributor.author: Mucalica, Ana
dc.description.abstract: In 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.
2021-01-01T00:00:00ZLorenz’s system - analysis of a sensitive systemLisitza, Cassandrahttps://hdl.handle.net/20.500.14078/22982023-02-10T22:31:02Z2021-01-01T00:00:00Zdc.title: Lorenz’s system - analysis of a sensitive system
dc.contributor.author: Lisitza, Cassandra
dc.description.abstract: Meteorology 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.
2021-01-01T00:00:00ZVan der Pol oscillator – analysis of a non-conservative systemReeves, Adamhttps://hdl.handle.net/20.500.14078/16432023-03-23T18:01:34Z2020-01-01T00:00:00Zdc.title: Van der Pol oscillator – analysis of a non-conservative system
dc.contributor.author: Reeves, Adam
dc.description.abstract: The 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.
2020-01-01T00:00:00Z