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Cohen-Macaulay Type Of Weighted Path Ideals, Shuai Wei
Cohen-Macaulay Type Of Weighted Path Ideals, Shuai Wei
All Dissertations
In this dissertation we give a combinatorial characterization of all the weighted $r$-path suspensions for which the $f$-weighted $r$-path ideal is Cohen-Macaulay. In particular, it is shown that the $f$-weighted $r$-path ideal of a weighted $r$-path suspension is Cohen-Macaulay if and only if it is unmixed. Type is an important invariant of a Cohen-Macaulay homogeneous ideal in a polynomial ring $R$ with coefficients in a field. We compute the type of $R/I$ when $I$ is any Cohen-Macaulay $f$-weighted $r$-path ideal of any weighted $r$-path suspension, for some chosen function $f$. In particular, this computes the type for all weighted trees …
Efficiency Of Homomorphic Encryption Schemes, Kyle Yates
Efficiency Of Homomorphic Encryption Schemes, Kyle Yates
All Theses
In 2009, Craig Gentry introduced the first fully homomorphic encryption scheme using bootstrapping. In the 13 years since, a large amount of research has gone into improving efficiency of homomorphic encryption schemes. This includes implementing leveled homomorphic encryption schemes for practical use, which are schemes that allow for some predetermined amount of additions and multiplications that can be performed on ciphertexts. These leveled schemes have been found to be very efficient in practice. In this thesis, we will discuss the efficiency of various homomorphic encryption schemes. In particular, we will see how to improve sizes of parameter choices in homomorphic …
On Complete Integral Closure Of Integral Domains, Todd Fenstermacher
On Complete Integral Closure Of Integral Domains, Todd Fenstermacher
All Dissertations
Given an integral domain D with quotient field K, an element x in K is called integral over D if x is a root of a monic polynomial with coefficients in D. The notion of integrality has roots in Dedekind's work with algebraic integers, and was later developed more rigorously by Emmy Noether. Different variations or generalizations of integrality have since been studied, including almost integrality and pseudo-integrality. In this work we give a brief history of integrality and almost integrality before developing the basic theory of these two notions. We will continue the theory of almost integrality further by …
Identifying Trace Affine Linear Sets Using Homotopy Continuation, Julianne Mckay
Identifying Trace Affine Linear Sets Using Homotopy Continuation, Julianne Mckay
All Theses
We investigate how the coefficients of a sparse polynomial system influence the sum, or the trace, of its solutions. We discuss an extension of the classical trace test in numerical algebraic geometry to sparse polynomial systems. Two known methods for identifying a trace affine linear subset of the support of a sparse polynomial system use sparse resultants and polyhedral geometry, respectively. We introduce a new approach which provides more precise classifications of trace affine linear sets than was previously known. For this new approach, we developed software in Macaulay2.
Minimal Differential Graded Algebra Resolutions Related To Certain Stanley-Reisner Rings, Todd Anthony Morra
Minimal Differential Graded Algebra Resolutions Related To Certain Stanley-Reisner Rings, Todd Anthony Morra
All Dissertations
We investigate algebra structures on resolutions of a special class of Cohen-Macaulay simplicial complexes. Given a simplicial complex, we define a pure simplicial complex called the purification. These complexes arise as a generalization of certain independence complexes and the resultant Stanley-Reisner rings have numerous desirable properties, e.g., they are Cohen-Macaulay. By realizing the purification in the context of work of D'alì, et al., we obtain a multi-graded, minimal free resolution of the Alexander dual ideal of the Stanley-Reisner ideal. We augment this in a standard way to obtain a resolution of the quotient ring, which is likewise minimal and multi-graded. …
Characterizing Unmixed Trees And Coronas With Respect To Pmu Covers, Michael Cowen
Characterizing Unmixed Trees And Coronas With Respect To Pmu Covers, Michael Cowen
All Dissertations
In this dissertation we study the algebraic properties of ideals constructed from graphs. We use algebraic techniques to study the PMU Placement Problem from electrical engineering which asks for optimal placement of sensors, called PMUs, in an electrical power system. Motivated by algebraic and geometric considerations, we characterize the trees for which all minimal PMU covers have the same size. Additionally, we investigate the power edge ideal of Moore, Rogers, and Sather-Wagstaff which identifies the PMU covers of a power system like the edge ideal of a graph identifies the vertex covers. We characterize the trees for which the power …
Conductors And Rings With Shared Ideals, Sydney Maibach
Conductors And Rings With Shared Ideals, Sydney Maibach
All Theses
Given an additive subgroup $I$ of a field $K$, we define the colon ideal (I:I) = {\alpha \in K: \alpha I \subseteq I}. We then use this to construct collections of rings with shared ideals and explore relationships between these concepts and the complete integral closure.
The Hfd Property In Orders Of A Number Field, Grant Moles
The Hfd Property In Orders Of A Number Field, Grant Moles
All Theses
We will examine orders R in a number field K. In particular, we will look at how the generalized class number of R relates to the class number of its integral closure R. We will then apply this to the case when K is a quadratic field to produce a more specific relation. After this, we will focus on orders R which are half-factorial domains (HFDs), in which the irreducible factorization of any element α∈R has fixed length. We will determine two cases in which R is an HFD if and only if its ring of …
Lyubeznik Ideals Minimally Generated By Four Or Fewer Elements, Nathan S. Fontes
Lyubeznik Ideals Minimally Generated By Four Or Fewer Elements, Nathan S. Fontes
All Theses
Free resolutions for an ideal are constructions that tell us useful information about the structure of the ideal. Every ideal has one minimal free resolution which tells us significantly more about the structure of the ideal. In this thesis, we consider a specific type of resolution, the Lyubeznik resolution, for a monomial ideal I, which is constructed using a total order on the minimal generating set G(I). An ideal is called Lyubeznik if some total order on G(I) produces a minimal Lyubeznik resolution for I. We investigate the problem of characterizing whether an ideal I is Lyubeznik …
Improved First-Order Techniques For Certain Classes Of Convex Optimization, Trevor Squires
Improved First-Order Techniques For Certain Classes Of Convex Optimization, Trevor Squires
All Dissertations
The primary concern of this thesis is to explore efficient first-order methods of computing approximate solutions to convex optimization problems. In recent years, these methods have become increasingly desirable as many problems in fields such as machine learning and imaging science have scaled tremendously. Our aim here is to acknowledge the capabilities of such methods and then propose new techniques that extend the reach or accelerate the performance of the existing state-of-the-art literature.
Our novel contributions are as follows. We first show that the popular Conditional Gradient Sliding (CGS) algorithm can be extended in application to objectives with H\"older continuous …