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Full-Text Articles in Logic and Foundations
A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu
A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu
HMC Senior Theses
The goal of this senior thesis is to explore general nonstandard analysis and some possible applications to 𝐶*-algebras in functional analysis. More specifically, we shall define an approximate identity of a 𝐶*-algebra using nonstandard analysis and study nonstandard hulls of internal 𝐶*-algebra in the context of different unitizations. We shall also prove a few results for ideals in 𝐶*-algebra using nonstandard definitions of approximate identities. We shall also briefly discuss the history and developments of nonstandard analysis.
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
HMC Senior Theses
Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …
Sudoku Variants On The Torus, Kira A. Wyld
Sudoku Variants On The Torus, Kira A. Wyld
HMC Senior Theses
This paper examines the mathematical properties of Sudoku puzzles defined on a Torus. We seek to answer the questions for these variants that have been explored for the traditional Sudoku. We do this process with two such embeddings. The end result of this paper is a deeper mathematical understanding of logic puzzles of this type, as well as a fun new puzzle which could be played.