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Physical Sciences and Mathematics Commons™
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Full-Text Articles in Physical Sciences and Mathematics
Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, Connor Hurley
Choice Of Choice: Paradoxical Results Surrounding Of The Axiom Of Choice, Connor Hurley
Honors Theses
When people think of mathematics they think "right or wrong," "empirically correct" or "empirically incorrect." Formalized logically valid arguments are one important step to achieving this definitive answer; however, what about the underlying assumptions to the argument? In the early 20th century, mathematicians set out to formalize these assumptions, which in mathematics are known as axioms. The most common of these axiomatic systems was the Zermelo-Fraenkel axioms. The standard axioms in this system were accepted by mathematicians as obvious, and deemed by some to be sufficiently powerful to prove all the intuitive theorems already known to mathematicians. However, this system …
Elliptic Curve Cryptology, Francis Rocco
Elliptic Curve Cryptology, Francis Rocco
Honors Theses
In today's digital age of conducting large portions of daily life over the Internet, privacy in communication is challenged extremely frequently and confidential information has become a valuable commodity. Even with the use of commonly employed encryption practices, private information is often revealed to attackers. This issue motivates the discussion of cryptology, the study of confidential transmissions over insecure channels, which is divided into two branches of cryptography and cryptanalysis. In this paper, we will first develop a foundation to understand cryptography and send confidential transmissions among mutual parties. Next, we will provide an expository analysis of elliptic curves and …
Efficient Denoising And Sharpening Of Color Images Through Numerical Solution Of Nonlinear Diffusion Equations, Linh T. Duong
Efficient Denoising And Sharpening Of Color Images Through Numerical Solution Of Nonlinear Diffusion Equations, Linh T. Duong
Honors Theses
The purpose of this project is to enhance color images through denoising and sharpening, two important branches of image processing, by mathematically modeling the images. Modifications are made to two existing nonlinear diffusion image processing models to adapt them to color images. This is done by treating the red, green, and blue (RGB) channels of color images independently, contrary to the conventional idea that the channels should not be treated independently. A new numerical method is needed to solve our models for high resolution images since current methods are impractical. To produce an efficient method, the solution is represented as …
Solving The Yang-Baxter Matrix Equation, Mallory O. Jennings
Solving The Yang-Baxter Matrix Equation, Mallory O. Jennings
Honors Theses
The Yang-Baxter equation is one that has been widely used and studied in areas such as statistical mechanics, braid groups, knot theory, and quantum mechanics. While many sets of solutions have been found for this equation, it is still an open problem. In this project, I solve the Yang-Baxter matrix equation that is similar in format to the Yang-Baxter equation. I try to solve the corresponding Yang-Baxter matrix equation, ������=������, where X is an unknown ������ matrix, and ��=[0����0] or [0−��−��0], by using the Jordan canonical form to find infinitely many solutions.