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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat
Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat
Honors Theses
Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this paper, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the torsion subgroups of elliptic curves. In particular, we will examine two methods used to classify these subgroups. We finish by employing these methods to categorize the torsion subgroups for a specific family of elliptic …
Category Theory And Universal Property, Niuniu Zhang
Category Theory And Universal Property, Niuniu Zhang
Honors Theses
Category theory unifies and formalizes the mathematical structure and concepts in a way that various areas of interest can be connected. For example, many have learned about the sets and its functions, the vector spaces and its linear transformation, and the group theories and its group homomorphism. Not to mention the similarity of structure in topological spaces, as the continuous function is its mapping. In sum, category theory represents the abstractions of other mathematical concepts. Hence, one could use category theory as a new language to define and simplify the existing mathematical concepts as the universal properties. The goal of …
Galois Theory And The Quintic Equation, Yunye Jiang
Galois Theory And The Quintic Equation, Yunye Jiang
Honors Theses
Most students know the quadratic formula for the solution of the general quadratic polynomial in terms of its coefficients. There are also similar formulas for solutions of the general cubic and quartic polynomials. In these three cases, the roots can be expressed in terms of the coefficients using only basic algebra and radicals. We then say that the general quadratic, cubic, and quartic polynomials are solvable by radicals. The question then becomes: Is the general quintic polynomial solvable by radicals? Abel was the first to prove that it is not. In turn, Galois provided a general method of determining when …
Introduction To Computational Topology Using Simplicial Persistent Homology, Jason Turner, Brenda Johnson, Ellen Gasparovic
Introduction To Computational Topology Using Simplicial Persistent Homology, Jason Turner, Brenda Johnson, Ellen Gasparovic
Honors Theses
The human mind has a natural talent for finding patterns and shapes in nature where there are none, such as constellations among the stars. Persistent homology serves as a mathematical tool for accomplishing the same task in a more formal setting, taking in a cloud of individual points and assembling them into a coherent continuous image. We present an introduction to computational topology as well as persistent homology, and use them to analyze configurations of BuckyBalls®, small magnetic balls commonly used as desk toys.
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
Honors Theses
Though it may seem non-intuitive, abstract algebra is often useful in the study of number theory. In this thesis, we explore some uses of abstract algebra to prove number theoretic statements. We begin by examining the structure of unique factorization domains in general. Then we introduce number fields and their rings of algebraic integers, whose structures have characteristics that are analogous to some of those of the rational numbers and the rational integers. Next we discuss quadratic fields, a special case of number fields that have important applications to number theoretic problems. We will use the structures that we introduce …
Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre
Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre
Honors Theses
It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that …