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Full-Text Articles in Set Theory
Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov
Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov
Electronic Theses, Projects, and Dissertations
Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig's lemma …
Symmetric Presentations And Related Topics, Mayra Mcgrath
Symmetric Presentations And Related Topics, Mayra Mcgrath
Electronic Theses, Projects, and Dissertations
In this thesis, we have investigated several permutation and monomialprogenitors for finite images. We have found original symmetric presen-tations for several important non-abelian simple groups, including lineargroups, unitary groups, alternating groups, and sporadic simple groups.We have found a number of finite images, including : L(2,41), PSL(2,11)×2, L(2,8), and L(2,19), as homomorphic images of the permutation progenitors. We have also found PGL(2,16) : 2 =Aut(PSL(2,16)) and PSL(2,16) as homomorphic images of monomial progenitors. We have performed manual double coset enumeration of finte images. In addition, we have given the isomorphism class of each image that we have discovered. Presentation for all …
Tutte-Equivalent Matroids, Maria Margarita Rocha
Tutte-Equivalent Matroids, Maria Margarita Rocha
Electronic Theses, Projects, and Dissertations
We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte …
Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, Nitish Mittal
Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, Nitish Mittal
Electronic Theses, Projects, and Dissertations
This project investigates the development of four different proofs of the law of quadratic reciprocity, in order to study the critical reasoning process that drives discovery in mathematics. We begin with an examination of the first proof of this law given by Gauss. We then describe Gauss’ fourth proof of this law based on Gauss sums, followed by a look at Eisenstein’s geometric simplification of Gauss’ third proof. Finally, we finish with an examination of one of the modern proofs of this theorem published in 1991 by Rousseau. Through this investigation we aim to analyze the different strategies used in …