The system of rational numbers can be extended to the real number system by several methods. In this paper, we shall extend the rational number system by means of rational nests of intervals, and develop the elementary properties of the real numbers obtained by this extension.

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The purpose of this thesis is to explore the Surreal Numbers from an elementary, con- structivist point of view, with the intention of introducing the numbers in a palatable way for a broad audience with minimal background in any specific mathematical field. Created from two recursive definitions, the Surreal Numbers form a class that contains a copy of the real numbers, transfinite ordinals, and infinitesimals, combinations of these, and in- finitely many numbers uniquely Surreal. Together with two binary operations, the surreal numbers form a field. The existence of the Surreal Numbers is proven, and the class is constructed from nothing, starting with the integers and dyadic rationals, continuing into the transfinite ordinals and the remaining real numbers, and culminating with the infinitesimals and uniquely surreal numbers. Several key concepts are proven regarding the ordering and containment properties of the numbers. The concept of a surreal continuum is introduced and demonstrated. The binary operations are explored and demonstrated, and field properties are proven, using many methods, including transfinite induction.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 123-125. / by Sy D. Friedman. / Ph.D.

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the Higman-Sims and Held groups. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

Ramsey theory has to do with order within disorder. This thesis studies two Ramsey numbers, R(3; 3) and R(3; 4), to see if they can provide insight into finding larger Ramsey numbers. The numbers are studied with the help of computer programs. In the second part of the thesis we try to create a coloring of K45 which lacks monochromatic K5 and where each vertex has an equal degree for both color of edges. The results from studying R(3; 3) and R(3; 4) fail to give any further insight into larger Ramsey numbers. Every coloring of K45 we produce contains a monochromatic K5.

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This report focuses on the discovery of Fermat numbers as well as the subsequent innovations in processes for finding factors of Fermat numbers. The property of the prime factors of Fermat numbers, as well as the connections between Fermat numbers and other areas of mathematics, is also discussed. / text

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