Physical Sciences and Mathematics Commons^™

Special Fiber Rings Of Certain Height Four Gorenstein Ideals, Jaree Hudson

Theses and Dissertations

Let S be a set of four variables, k a field of characteristic not equal to two such that k contains all square roots, and I a height four Gorenstein ideal of k[S] generated by nine quadratics so that I has a Gorenstein-linear resolution. We define a complex X• so that each module of X• is the tensor product of a certain polynomial ring Q in nine variables and a direct sum of indecomposable k[Sym(S)]-modules and the differential maps are Q- and k[Sym(S)]-module homomorphisms. Work with the Macaulay2 software suggests that H0(X•) is the special fiber ring of I and …

Turán Problems And Spectral Theory On Hypergraphs And Tensors, Shuliang Bai

Theses and Dissertations

Turán problems on uniform hypergraphs have been actively studied for many decades. However, on non-uniform hypergraphs, these problems are rarely considered. We refer a non-uniform hypergraph as an R-hypergraph where R is the set of cardinalities of all edges. An R-graph H is called degenerate if it has the smallest Turán density |R(H)|−1. What do the degenerate R-graphs look like? For the special case R = {r}, the answer to this question is simple: they are r-partite r-uniform hypergraphs. However, it is more intrigue for the other cases. A degenerate hypergraph is called trivial if it is contained in the …

A Quest For Positive Definite Matrices Over Finite Fields, Erin Patricia Hanna

Theses and Dissertations

Positive definite matrices make up an interesting and extremely useful subset of Hermitian matrices. They are particularly useful in exploring convex functions and finding minima for functions in multiple variables. These matrices admit a plethora of equivalent statements and properties, one of which is an existence of a unique Cholesky decomposition. Positive definite matrices are not usually considered over finite fields as some of the definitions and equivalences are quickly seen to no longer hold. Motivated by a result from the theory of pressing sequences, which almost mirrors an equivalent statement for positive definite Hermitian matrices, we consider whether any …

Geometry Of Derived Categories On Noncommutative Projective Schemes, Blake Alexander Farman

Theses and Dissertations

Noncommutative Projective Schemes were introduced by Michael Artin and J.J. Zhang in their 1994 paper of the same name as a generalization of projective schemes to the setting of not necessarily commutative algebras over a commutative ring. In this work, we study the derived category of quasi-coherent sheaves associated to a noncommutative projective scheme with a primary emphasis on the triangulated equivalences between two such categories.

We adapt Artin and Zhang’s noncommutative projective schemes for the language of differential graded categories and work in Ho (dgcatk), the homotopy category of differential graded categories, making extensive use of Bertrand Toën’s Derived …

Quick Trips: On The Oriented Diameter Of Graphs, Garner Paul Cochran

Theses and Dissertations

In this dissertation, I will discuss two results on the oriented diameter of graphs with certain properties. In the first problem, I studied the oriented diameter of a graph G. Erdos et al. in 1989 showed that for any graph with |V | = n and δ(G) = δ the maximum the diameter could possibly be was 3 n/ δ+1. I considered whether there exists an orientation on a given graph with |G| = n and δ(G) = δ that has a small diameter. Bau and Dankelmann (2015) showed that there is an orientation of diameter 11 n/ δ+1 + …

Local Rings And Golod Homomorphisms, Thomas Schnibben

Theses and Dissertations

The Poincaré series of a local ring is the generating function of the Betti numbers for the residue field. The question of when this series represents a rational function is a classical problem in commutative algebra. Golod rings were introduced by Golod in 1962 and are one example of a class of rings that have rational Poincaré series. The idea was generalized to Golod homomorphisms by Levin in 1975.

In this paper we prove two homomorphisms are Golod. The first is a class of ideals such that the natural projection to the quotient ring is a Golod homomorphism. The second …