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Full-Text Articles in Algebra
Harmony Amid Chaos, Drew Schaffner
Harmony Amid Chaos, Drew Schaffner
Pence-Boyce STEM Student Scholarship
We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as …
Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj
Algebraic And Geometric Properties Of Hierarchical Models, Aida Maraj
Theses and Dissertations--Mathematics
In this dissertation filtrations of ideals arising from hierarchical models in statistics related by a group action are are studied. These filtrations lead to ideals in polynomial rings in infinitely many variables, which require innovative tools. Regular languages and finite automata are used to prove and explicitly compute the rationality of some multivariate power series that record important quantitative information about the ideals. Some work regarding Markov bases for non-reducible models is shown, together with advances in the polyhedral geometry of binary hierarchical models.