Mathematics and Statistics MRP
http://hdl.handle.net/10464/14573
Fri, 23 Feb 2024 22:38:43 GMT2024-02-23T22:38:43ZSome Families of Elliptic Curves
http://hdl.handle.net/10464/18169
Some Families of Elliptic Curves
Shah, Sudev
Elliptic 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.
http://hdl.handle.net/10464/18169A Study of Soccer Space Gain in Pass Sequences using Logistic Regression
http://hdl.handle.net/10464/17929
A Study of Soccer Space Gain in Pass Sequences using Logistic Regression
Salim, Christopher
Some 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.
http://hdl.handle.net/10464/17929Machine Learning Approaches for Estimating Prevalence of Undiagnosed Hypertension among Bangladeshi Adults: Evidence from a Nationwide Survey
http://hdl.handle.net/10464/17629
Machine Learning Approaches for Estimating Prevalence of Undiagnosed Hypertension among Bangladeshi Adults: Evidence from a Nationwide Survey
Siddiquee, Tanjim
In 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.
http://hdl.handle.net/10464/17629Diophantine Equation in Logarithms
http://hdl.handle.net/10464/17555
Diophantine Equation in Logarithms
Tian, Zhao
The 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 .
http://hdl.handle.net/10464/17555