Mathematics and Statistics MRP
Permanent URI for this collectionhttps://hdl.handle.net/10464/14573
Students currently enrolled in the Mathematics and Statistics graduate program here at Brock University will be required to submit an electronic copy of their final Major Research Paper to this repository as part of graduation requirements. Instructions on how to do this can be found online
Once your MRP has been accepted in the Repository you will receive an email confirmation along with a link to your workBrowse
Recent Submissions
Item Open Access Exploring Computational Thinking and Mathematical Reasoning Through Programming-Based Mathematics Tasks in Scratch and PythonSinaei, MaryamThis exploratory study seeks to better understand computational thinking (CT), mathematical reasoning (MR), and their interplay in solving programming-based mathematical tasks at the secondary level. Theoretically, I mainly adopt Kallia et al.’s (2021) characterization of CT for mathematics education and Jeannotte and Kieran’s (2017) model of MR. Empirically, I reflect on my engagement with three mathematical tasks drawn from Gadanidis’ (2022) Math + Coding Teams, using Scratch and Python. The analysis consists of identifying CT components (e.g., abstraction, decomposition, algorithmic thinking) and MR processes (e.g., conjecturing, generalizing, justifying), and their interaction. My analysis suggests that CT, through coding, supported my problem-solving and enabled hypothesis testing and iterative MR refinement. In other words, the programming environments served not only as execution tools but as spaces for mathematics inquiry and exploration, enhancing both my CT and MR engagement. These findings highlight the value of integrating coding into mathematics education to foster conceptual reasoning. Keywords: Computational Thinking, Mathematical Reasoning, Programming-Based Tasks, Mathematics Education, Scratch and Python.Item Open Access Advancing Generative Modelling and Applications with Boltzmann Machines, Restricted Boltzmann Machines, and Sum-Product NetworksAhmad, Mo AhsanIn the era of advanced machine learning methodologies, generative probabilistic modelling shows great promise for solving real-world problems. This study focuses on Boltzmann Machines (BMs), Restricted Boltzmann Machines (RBMs), and Sum-Product Networks (SPNs), highlighting their abilities to reconstruct complex data distributions and produce meaningful outputs. Notably, BMs and RBMs excel at modelling data distributions, while SPNs utilize hierarchical structures for efficient representation and scalable probabilistic inference. Using the Fashion MNIST dataset as a benchmark, this work demonstrates the models' practicality through reconstructed images, precise predictions, and performance metrics. The findings confirm their applicability in tasks such as image generation, object recognition, and pattern matching. This study provides an empirical assessment of the strengths and limitations of each approach while expanding the potential applications of generative models in machine learning.Item Open Access Optimal Design and Planning for Random Effect Models and Models with Measurement Errors.SARFO, BRIDGET KONADUThis project addresses optimal design problems for linear mixed models with covariates (mixed ANCOVA models), and linear fixed models with measurement errors. First, we examine ANCOVA mixed models, focusing on efficient treatment allocations. We develop optimal designs using a general framework based on D-optimality, proposing a two-stage design approach to account for unknown parameters. In this approach, variances of random effects across treatment groups are considered distinct. Second, we formulate A-optimal designs for measurement error models, addressing errors in the response alone or in both response and explanatory variables. We propose that optimal designs for models with measurement errors significantly outperform classical designs. Furthermore, our results in A-optimal design for the models with measurement errors in both explanatory and response variables provide significantly greater efficiencies compared to classical designs.Item Open Access A Search for Integrable Evolution Equations with Lax Pairs over the OctonionsLAM, Hin HoLax pairs play a vital role in the integrability theory of evolution equations because they are used for the inverse scattering transformation to generate multi-soliton solutions. As an extension of integrable real evolution equations and their known Lax pair, this report focuses on the search for octonion evolution equations u_t = F(u, u_x, u_xx, u_xxx) of KdV type and mKdV type that have a Lax pair, where u(t, x) is an octonion variable. A Lax pair is defined as L_t ψ = M(Lψ) − L(Mψ) with linear differential operators L and M whose coefficients depend on u and x-derivatives of u, where ψ(t, x) is an auxiliary octonion function. These operators act on ψ by producing a linear polynomial in ψ and x-derivatives of ψ such that each term is a product involving u, u_x, . . . , and (x-derivatives of) ψ in a given order. It is assumed that the evolution equation u_t = F(u, u_x, u_xx, u_xxx) as well as both Lψ and Mψ are homogeneous under a scaling of t, x, u which is either the scaling associated to the KdV equation or the mKdV equation. This leads to an overdetermined system of algebraic equations for the (real-valued) coefficients of u and x−derivatives of u in F, Lψ and Mψ. The formulation of the overdetermined system involves two important differences compared to the case of a real variable u. Firstly, since octonions are non-associative and non-commutative, F, Lψ and Mψ contain many more terms, with different orderings of products. In particular, ψ (and its x-derivatives) are allowed to appear on the left, in the middle, or on the right. Secondly, products of octonions obey certain algebraic identities, whereby terms that are equivalent modulo these identities must be eliminated. To solve the overdetermined system, Maple is used to do the splittings, and depending on the complexity of the system, ’rifsimp’ in Maple or a package called ’Crack’ in Reduce are used to solve it. As a main result, a single KdV octonion equation, three mKdV octonion equations, and also a single potential-KdV octonion equation, each of which has more than one Lax pair, are obtained.Item Open Access The Impact of Exchange Rate Data on Canadian Inflation: An FPCA and Group LASSO ApproachTasnim, SamihaThis study investigates the temporal dynamics of exchange rates between various international currencies and the Canadian dollar, with a focus on understanding how these rates influence Canadian inflation. The Functional Principal Component Analysis (FPCA) is applied to effectively reduce the dimensionality of exchange rate data and capture important modes of variation. The extracted functional principal components (FPCs) were then used in a Group LASSO regression model to identify which currencies most significantly impact inflation rates in Canada. Our analysis includes exchange rates from nine countries. The results show that the U.S. Dollar (USD), Mexican Peso (MXN), and Swedish Krona (SEK) are the most influential currencies in predicting Canadian inflation rates. By employing these advanced statistical techniques, this study provides a comprehensive assessment of how fluctuations in global currencies can affect the domestic economic environment, offering valuable insights for policymakers and financial analysts. This study contributes to the broader understanding of currency exchange impacts on inflation and highlights the importance of specific international currencies in economic forecasting.Item Open Access Integrability of Evolutionary Type EquationsXU, XINDIClassical Integrable Systems represent a captivating and profound branch of mathematical physics, offering a framework to understand the motion of particles in a way that goes beyond mere mathematical description. These systems possess a remarkable property known as complete integrability, meaning that they admit a sufficient number of independent constants of motion, allowing for their trajectories to be precisely determined. The development of this field gained momentum in the 18th and 19th centuries, marked by the discovery of various integrable systems associated with fundamental equations of motion. In mathematical physics, CIS refer to a special class of differential equations that possess unique characteristics. In this research project, we will begin with the introduction of several important and fundamental integrable systems, such as the Hopf equation, Burgers equation, and Liouville equation.Item Open Access A Review of Life Insurance’s Secondary MarketGuo, ChuyunThe secondary market for life insurance policies emerged to provide financial relief for terminally ill patients through viatical settlements. Over time, it expanded to include life settlements for seniors and individuals with chronic conditions. This growth was driven by demographic shifts and increased life expectancies. This paper discusses the historical evolution of the secondary market, the types of transactions involved, and the market dynamics. It explores the benefits and challenges faced by policyholders, insurance companies, and investors. For policyholders, the market offers enhanced liquidity and fair compensation. However, it also requires insurers to adjust premium structures and manage adverse selection. Investors benefit from high returns and risk diversification, but they are also exposed to longevity risk, liquidity risk, return volatility, and regulatory changes. The study highlights the complexities of asymmetric information and emphasizes the need for sophisticated actuarial models and robust regulatory frameworks to ensure market stability and sustainability.Item Open Access Review of Lapse and Reentry Behavior and Its Impact on the Design of Variable AnnuitiesZhao, SiqinThis project explores the role of Variable Annuities (VAs) and Registered Index-Linked Annuities (RILAs) within contemporary financial planning, emphasizing how they adapt to evolving market conditions. A comprehensive analysis of these products is provided, including their protective mechanisms, pricing models, and risk characteristics, to explain why they effectively meet investor needs. The evolution from traditional annuities to VAs is traced, and the further development of RILAs from VAs is discussed. The study particularly focuses on the behavior of policyholders who often choose to surrender their policies for more favorable options, thereby maximizing their benefits. Through qualitative and quantitative analysis, the paper illustrates how these products respond to market developments.Item Open Access Some Families of Elliptic CurvesShah, SudevElliptic curves, intricate mathematical structures, form a nexus between number theory, alge- braic geometry, and cryptography. This paper offers a thorough exploration of these curves, delving into their foundational properties, historical origins, and diverse applications. Beginning with an introduction to the basics of elliptic curves, including their Weierstrass form, group theory, and fundamental concepts such as the group law and torsion points, the paper traces the historical evolution of elliptic curve theory, recognizing the contributions of mathematicians like Abel, Jacobi, and Weierstrass. The crux of the paper by G. Walsh lies in extending prior research by effectively proving that for sufficiently large values of m, elliptic curves expressed as y^2 = f(x) + m^2, where f(x) is a cubic polynomial splitting over the integers, have a rank of at least 2. This result stands as an effective version of Shioda’s theorem, marking a significant advancement in the field. Moreover, the paper delves into the pivotal role of elliptic curve cryptography (ECC) in modern secure communication systems. ECC provides robust encryption, digital signatures, and key exchange protocols, leveraging the security and efficiency advantages inherent in elliptic curves. The paper emphasizes ECC’s prominence in contemporary cryptography, illustrating its preference in securing digital data transmission. Additionally, the paper explores recent developments, including endeavours to address the Birch and Swinnerton-Dyer conjecture. It also highlights the relevance of elliptic curves in solving complex mathematical problems, such as Diophantine equations and Fermat’s Last Theorem, underscoring their broader significance in number theory. In essence, this paper serves as a comprehensive guide to elliptic curves, illuminating their mathematical elegance and practical utility. It underscores their indispensable role in modern cryptography while acknowledging their enduring impact on the realm of mathematics. By unravelling the theoretical intricacies and real-world applications of elliptic curves, this paper invites readers to appreciate the profound interconnection between pure mathematical concepts and their transformative influence on contemporary technology.Item Open Access A Study of Soccer Space Gain in Pass Sequences using Logistic RegressionSalim, ChristopherSome pass sequences open up more space on the pitch than others. Several studies about space gain in soccer have been performed in the past, but the relationship between space gain and the ability to score a goal has not been established yet. This research aims to predict goal occurrence by using total space gain for each pass sequence as the explanatory variable in logistic regression. Combining event and tracking data from the 2019 regular season of Chinese Super League (CSL), space quality can be calculated. We implemented space quality calculation from for 237 matches in the 2019 CSL dataset. Space quality is defined as the product of likelihoods that a team can gain control at a given location and time and the defending team can impede scoring attempts. The research demonstrated that for every unit the total space gain increased in a given pass sequence, the scoring odds increased by 23%. This finding showcases that pass sequences that create space are more likely to help in scoring, which is consistent with real-life soccer events. Combining the space occupation gain and probability of scoring a goal in every pass sequence, a team could make an informed decision of how they should position each player in a given scenario.Item Open Access Machine Learning Approaches for Estimating Prevalence of Undiagnosed Hypertension among Bangladeshi Adults: Evidence from a Nationwide SurveySiddiquee, TanjimIn South Asia, hypertension is the most prevalent modifiable risk factor for cardiovascular disorders. Comparing machine learning to statistical approaches, it has been found that it performs better at identifying clinical risk. This study utilized machine learning techniques to estimate undiagnosed hypertension. We created a single dataset out of individual-level data from the Bangladesh Demographic and Health Survey (2017-18). The JNC-7 and ACCAHA criteria were used to define hypertension. We used two well-known ML approaches logistic regression and log-binomial regression to determine the prevalence of undiagnosed hypertension. A considerable number (16%) of hypertension cases in Bangladesh are still undiagnosed. Young people and the divisions of Sylhet and Rangpur were found to be more at risk for undetected hypertension. ML models performed well at identifying undiagnosed hypertension and its contributing factors in South Asia. Future studies incorporating biological markers will be necessary to improve the ML algorithms and determine their applicability.Item Open Access Diophantine Equation in LogarithmsTian, ZhaoThe main work of these pages is written by myself under the supervisor of Dr. Omar Kihel, pertaining to continued fractions and applications , linear form in logarithms and the solutions of Diophantine equation Fn1 + Fn2 + Fn3 + Fn4 = 6a . The initial aim of the paper was to explore the possible solutions of the Diophantine equations in the form of Fn1 +Fn2 +Fn3 +Fn4 = y a . I begin my thesis by establishing some preliminary results and applications. The paper managed to extend the ideas of results of the Diophantine equations Fn1 +Fn2 +Fn3 +Fn4 = 2a and Fn1 +Fn2 +Fn3 +Fn4 = 11a . Mattveev Theorem, Legendre Theorem and a lemma by Dujella-petho are key theorems which we establish the main result. This paper includes the result of Diophantine equation Fn1 +Fn2 +Fn3 +Fn4 = 6a and it may require computations by computers. I will begin by introducing continued fractions, leading to linear forms in logarithms, followed by a section on the necessary preliminaries on Fibonacci numbers which concludes my results of the sum of four Fibonacci numbers. I then move to explore the aforementioned solutions of Fn1 + Fn2 + Fn3 + Fn4 = 6a .Item Open Access Permutation Polynomials over Finite Fields and their application to CryptographyBenseba, KatiaThe aim of the paper is the study of Permutation Polynomials over finite fields and their application to cryptography. In this paper, I will begin by a brief review of finite fields, define permutation polynomials over finite fields and their properties. I will present old results such as Hermite-Dickson’s Theorem as well as some most recent ones. After introducing cryptog- raphy, I will give a historical overview, by explaining some cryptosystems such as RSA and ElGamal. Finally, I will present some cryptographical protocols based on Permutation Polynomials over Finite Fields.