Mathematics Commons^™

Computing The Canonical Ring Of Certain Stacks, Jesse Franklin

Graduate College Dissertations and Theses

We compute the canonical ring of some stacks. We first give a detailed account of what this problem means including several proofs of a famous historical example. The main body of work of this thesis expands on our article \cite{Franklin-geometry-Drinfeld-modular-forms} in describing the geometry of Drinfeld modular forms as sections of a specified line bundle on a certain stacky modular curve. As a consequence of that geometry, we provide a program: one can compute the (log) canonical ring of a stacky curve to determine generators and relations for an algebra of Drinfeld modular forms, answering a problem posed by Gekeler …

Ring Learning With Errors, Sarah Days-Merrill

Graduate College Dissertations and Theses

Over the last twenty years, lattice-based cryptosystems have gained interest due to their levelof security against attacks from quantum computers. The main cryptosystems are based on the hardness of Ring Learning with Errors (RLWE). The Learning with Errors (LWE) problems were first introduced in 2005 by Regev [Reg09] and in 2010, [LPR10] developed the Ring Learning with Errors (RLWE) problems as candidates for safe encryption against quantum computers. Let K be a number field with ring of integers OK. For a prime q, the RLWE problems rely on samples of the form (a, b) ∈ OK/qOK × OK/qOK where a …

Edge Colored And Edge Ordered Graphs, Per Gustin Wagenius

Graduate College Dissertations and Theses

In this work, the current state of the field of edge-colored graphs is surveyed. Anew concept of unshrinkable edge colorings is introduced which is useful for rainbow subgraph problems and interesting in its own right. This concept is analyzed in some depth. Building upon the linear edge ordering described in a recent work from Gerbner, Methuku, Nagy, Pálvölgyi, Tardos, and Vizer, edge-ordering graphs with the cyclic group is introduced and some results are given on this and a related counting problem.

Data-Driven Reachability Of Non-Linear Systems Via Optimization Of Chen-Fliess Series, Ivan Perez Avellaneda

Graduate College Dissertations and Theses

A reachable set is the set of all possible states produced by applying a set of inputs, initial states, and parameters. The fundamental problem of reachability is checking if a set of states is reached provided a set of inputs, initial states, and parameters, typically, in a finite time. In the engineering field, reachability analysis is used to test the guarantees of the operation’s safety of a system. In the present work, the reachability analysis of nonlinear control affine systems is studied by means of the Chen-Fliess series. Different perspectives for addressing the reachability problem, such as interval arithmetic, mixed-monotonicity, …

Applications Of Centrality Measures And Extremal Combinatorics, Hunter Dane Rehm

Graduate College Dissertations and Theses

Centrality measures assign numbers or rankings to network nodes that reflect their importance. There are many types of centrality measures, each suitable for different types of networks and applications. In Chapter 2, we consider a model of astronaut health during a space mission. Katz centrality is commonly used to measure the influence of nodes in social and biological networks. We motivate its use in this application to estimate the expected quality time lost due to the progression of medical conditions. In Chapter 3, we find dominating sets in satellite networks. To do this, we use the Shapley value, a centrality …

An Analysis Of A Linear Algebra Based Group Key Exchange Protocol, Annie Zhang

Graduate College Dissertations and Theses

Group key exchange protocols are used to establish session keys, which can then be used as encryption keys to set up secure channels of communication, between more than two parties simultaneously. Many different group key exchange protocols exist and require security proofs in order to determine the strength of the protocol and answer the following questions: does the protocol provide authentication, and if so, to what degree? Does the protocol provide key secrecy? In this thesis we examine a particular group key exchange protocol that we call the \textit{vector space projection protocol} as first described in “A Group Key Establishment …

Studies On The Tempered Fractional Natural Decomposition Method, Nazek Ahmad Obeidat

Graduate College Dissertations and Theses

In Fractional Calculus (FC), the notion of fractional derivatives and integrals arise from convolutions with a power law, which, when multiplied by an exponential factor, one obtains tempered fractional derivatives and integrals.

The purpose of this dissertation is to develop theories and applications of a new technique in FC called the Tempered Fractional Natural Transform Method (TFNTM). This method can be used to solve a myriad of problems in linear and nonlinear ordinary and partial differential equations.

We present convergence analysis, give error estimates, and provide exact solutions, with graphical illustrations, to many well-known problems in tempered fractional differential equation, …

High-Resolution Downscaling And Bias-Correction Of Temperature And Precipitation: Advances In Statistical Methods, Maike Holthuijzen

Graduate College Dissertations and Theses

High-resolution, bias-corrected climate data is necessary for climate impact studies and modeling efforts at local scales. General circulation models (GCMs) provide important information about historical and future larger-scale climate trends, but their spatial resolution is too coarse to investigate localized effects of climate processes. Additionally, raw GCM output is characterized by some degree of bias. Two post-processing procedures known as downscaling and bias-correction are typically applied to raw climate model output prior to its use in further modeling applications. Downscaling is the process in which data at a coarse spatial scale is transformed to a fine spatial scale. Bias-correction refers …

Leveraging Behavioral Data For Public Health: Exploring Sleep And Nature Exposure Using Mobile Phones And Social Media, Kelsey Linnell

Graduate College Dissertations and Theses

Health surveillance and assessment are considered essential components of a func- tional public health system. The recent ubiquity of mobile devices and social media have created a wealth of behavioral data, and bring into existence new forms of pop- ulation health monitoring. These new digital sources can provide direct and passive data for more detailed and nuanced health factors, and have expanded the human, spatial, and temporal scales at which these factors can be measured. In this project, I leverage digital trace data from tweets and mobile device location pings to explore population scale sleep loss, and nature exposure through …

Quantifying Proverb Dynamics In Books, News Articles, And Tweets, Ethan Davis

Graduate College Dissertations and Theses

Proverbs are an essential component of language and culture, and though much attention has been paid to their history and currency, there has been comparatively little quantitative work on the frequency with which they are used, and the dynamics of their use over time. With wider availability of large corpora reflecting many diverse genres of documents, it is now possible to take a wider view of the importance of the proverb. Can a corpus linguistic approach to phraseology support existing histories, and what further insight can be gained from a quantitative approach? This study measures temporal changes in the relevance …

Loss Of Precision In Implementations Of The Toom-Cook Algorithm, Marcus Elia

Graduate College Dissertations and Theses

Historically, polynomial multiplication has required a quadratic number of operations. Several algorithms in the past century have improved upon this. In this work, we focus on the Toom-Cook algorithm. Devised by Toom in 1963, it is a family of algorithms parameterized by an integer, n. The algorithm multiplies two polynomials by recursively dividing them into smaller polynomials, multiplying many small polynomials, and interpolating to obtain the product. While it is no longer the asymptotically fastest method of multiplying, there is a range of intermediate degrees (typically less than 1000) where it performs the best.

Some applications, like quantum-resistant cryptosystems, require …

Jacobians Of Finite And Infinite Voltage Covers Of Graphs, Sophia Rose Gonet

Graduate College Dissertations and Theses

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a graph X; it is a finite abelian group whose cardinality is equal to the number of spanning trees of X (Kirchhoff's Matrix Tree Theorem). This dissertation proves results about the Jacobians of certain families of covering graphs, Y, of a base graph X, that are constructed from an assignment of elements from a group G to the edges of X (G is called the voltage group and Y is called the derived graph). The principal aim is to relate the Jacobian of …

Reference Governors For Time-Varying Systems And Constraints, Collin Freiheit

Graduate College Dissertations and Theses

Control systems are often subject to constraints imposed by physical limitations or safety considerations, and require means of constraint management to ensure the stability and safety of the system. For real-time implementation, constraint management schemes must not carry a heavy computational burden; however many of the current solutions are computationally unattractive, especially those with robust formulations. Thus, the design of constraint management schemes with low computational loads is an important and practical problem for control engineers. Reference Governor (RG) is an efficient constraint management scheme that is attractive for real-time implementation due to its low computational complexity and ease of …

Extremal/Saturation Numbers For Guessing Numbers Of Undirected Graphs, Jo Ryder Martin

Graduate College Dissertations and Theses

Hat guessing games—logic puzzles where a group of players must try to guess the color of their own hat—have been a fun party game for decades but have become of academic interest to mathematicians and computer scientists in the past 20 years. In 2006, Søren Riis, a computer scientist, introduced a new variant of the hat guessing game as well as an associated graph invariant, the guessing number, that has applications to network coding and circuit complexity. In this thesis, to better understand the nature of the guessing number of undirected graphs we apply the concept of saturation to guessing …

The Circuit And Cocircuit Lattices Of A Regular Matroid, Patrick Mullins

Graduate College Dissertations and Theses

A matroid abstracts the notions of dependence common to linear algebra, graph theory, and geometry. We show the equivalence of some of the various axiom systems which define a matroid and examine the concepts of matroid minors and duality before moving on to those matroids which can be represented by a matrix over any field, known as regular matroids. Placing an orientation on a regular matroid allows us to define certain lattices (discrete groups) associated to the matroid. These allow us to construct the Jacobian group of a regular matroid analogous to the Jacobian group of a graph. We then …

Some Results On A Set Of Data Driven Stochastic Wildfire Models, Maxfield E. Green

Graduate College Dissertations and Theses

Across the globe, the frequency and size of wildfire events are increasing. Research focused on minimizing wildfire is critically needed to mitigate impending humanitarian and environmental crises. Real-time wildfire response is dependent on timely and accurate prediction of dynamic wildfire fronts. Current models used to inform decisions made by the U.S. Forest Service, such as Farsite, FlamMap and Behave do not incorporate modern remotely sensed wildfire records and are typically deterministic, making uncertainty calculations difficult. In this research, we tested two methods that combine artificial intelligence with remote sensing data. First, a stochastic cellular automata that learns algebraic expressions was …

On The Dynamics And Structure Of Multiple Strain Epidemic Models And Genotype Networks, Blake Joseph Mitchell Williams

Graduate College Dissertations and Theses

Mathematical disease modeling has long operated under the assumption that any one infectious disease is caused by one transmissible pathogen. This paradigm has been useful in simplifying the biological reality of epidemics and has allowed the modeling community to focus on the complexity of other factors such as contact structure and interventions. However, there is an increasing amount of evidence that the strain diversity of pathogens, and their interplay with the host immune system, can play a large role in shaping the dynamics of epidemics.

This body of work first explores the role of strain-transcending immunity in mathematical disease models, …

Higher-Order Runge--Kutta Type Schemes Based On The Method Of Characteristics For Hyperbolic Equations With Crossing Characteristics, Jeffrey Steven Jewell

Graduate College Dissertations and Theses

The Method of Characteristics (MoC) is a well-known procedure used to find the numerical solution of systems of hyperbolic partial differential equations (PDEs). The main idea of the MoC is to integrate a system of ordinary differential equations (ODEs) along the characteristic curves admitted by the PDEs. In principle, this can be done by any appropriate numerical method for ODEs. In this thesis, we will examine the MoC applied to systems of hyperbolic PDEs with straight-line and crossing characteristics. So far, only first- and second-order accurate explicit MoC schemes for these types of systems have been reported. As such, the …

Mathematical Analysis Of Some Partial Differential Equations With Applications, Kewang Chen

Graduate College Dissertations and Theses

In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation …

Quadrature-Based Gravity Models For The Homogeneous Polyhedron, Jason Pearl

Graduate College Dissertations and Theses

A number of missions to comets and asteroids have been undertaken by major space organizations driving a need to accurately characterize their gravitational fields. This is complicated however by their irregular shapes. To accurately and safely navigate spacecraft in these environments, a simple point-mass gravity model is insufficient and instead higher-fidelity models are required. Several such models exist for this purpose but all posess drawbacks. Moreover, there are some applications for which the currently available models are not particular well suited.

In this dissertation, numerical quadrature and curvilinear meshing techniques are applied to the small body gravity problem. The goal …

Characterizing And Quantifying Vaccine Discourse Patterns On Social Media Amidst Global Pandemics, Amelia Tarren

Graduate College Dissertations and Theses

Globally, the 2019 Coronavirus pandemic (COVID-19) impacted and threatened the everyday lives and wellbeing of humanity. The development of a vaccine resulted in widespread discourse on social media platforms like Twitter, leading to potentially divergent sentiments about the COVID-19 vaccine. Given the novelty and polarity of vaccine sentiment and discourse, critical knowledge gaps exist as to how these factors develop. This study aims to characterize rapidly growing vaccine-specific dis- course by identifying words and phrases connected to the anchor 1-gram "vaccin*", the derivative of "vaccination" and "vaccinate", amongst others. This study draws from a collection of social media posts on …

Market Efficiency In U.S. Stock Markets: A Study Of The Dow 30 And The S&P 30, Colin Michael Van Oort

Graduate College Dissertations and Theses

The U.S. National Market System (NMS), the largest marketplace in the world for securities and exchange traded funds, suffers from geographic market fragmentation which leads to reduced market efficiency.

Communication lines transmit price updates and other information between geographically isolated exchanges at varying speeds, bounded above by the speed of light.

Market participants have access to federally mandated information provided by the Securities Information Processor (SIP) and privately offered information provided by the exchanges, often called direct feeds.

These feeds are quantitatively and qualitatively distinct, with the direct feeds tending to provide more information at a faster rate than the …

Networks, (K)Nots, Nucleotides, And Nanostructures, Ada Morse

Graduate College Dissertations and Theses

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of …

Some Results On A Class Of Functional Optimization Problems, David Rushing Dewhurst

Graduate College Dissertations and Theses

We first describe a general class of optimization problems that describe many natu- ral, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances of these problems in statistical physics, facility allocation, and machine learning. A dynamic description and statement of a partial inverse problem follow. When attempting to optimize the state of a system governed by the generalized equipartitioning princi- ple, it is vital to understand the nature of the governing probability distribution. We show that optimiziation for the incorrect probability distribution can have catas- trophic results, e.g., …

Novelty Detection Of Machinery Using A Non-Parametric Machine Learning Approach, Enrique Angola

Graduate College Dissertations and Theses

A novelty detection algorithm inspired by human audio pattern recognition is conceptualized and experimentally tested. This anomaly detection technique can be used to monitor the health of a machine or could also be coupled with a current state of the art system to enhance its fault detection capabilities. Time-domain data obtained from a microphone is processed by applying a short-time FFT, which returns time-frequency patterns. Such patterns are fed to a machine learning algorithm, which is designed to detect novel signals and identify windows in the frequency domain where such novelties occur. The algorithm presented in this paper uses one-dimensional …

Regularization In Symbolic Regression By An Additional Fitness Objective, Ryan Grindle

Graduate College Dissertations and Theses

Symbolic regression is a method for discovering functions that minimize error on a given dataset. It is of interest to prevent overfitting in symbolic regression. In this work, regularization of symbolic regression is attempted by incorporating an additional fitness objective. This new fitness objective is called Worst Neighbors (WN) score, which measures differences in approximate derivatives in the form of angles. To compute the Worst Neighbors score, place partition points between each pair of adjacent data points. For each pair of data points, compute the maximum angle between the line formed by the pair of data points and the lines …

An Exposition Of Selberg's Sieve, Jack Dalton

Graduate College Dissertations and Theses

A number of exciting recent developments in the field of sieve theory have been done concerning bounded gaps between prime numbers. One of the main techniques used in these papers is a modified version of Selberg's Sieve from the 1940's. While there are a number of sources that explain the original sieve, most, if not all, are quite inaccessible to those without significant experience in analytic number theory. The goal of this exposition is to change that. The statement and proof of the general form of Selberg's sieve is, by itself, difficult to understand and appreciate. For this reason, the …

Wronskian And Gram Solutions To Integrable Equations Using Bilinear Methods, Benjamin Wiggins

Graduate College Dissertations and Theses

This thesis presents Wronskian and Gram solutions to both the Korteweg-de Vries and Kadomtsev-Petviashvili equations, which are then scalable to arbitrarily large numbers of interacting solitons.

Through variable transformation and use of the Hirota derivative, these nonlinear partial differential equations can be expressed in bilinear form. We present both Wronskian and Gram determinants which satisfy the equations.

N=1,2,3 and higher order solutions are presented graphically; parameter tuning and the resultant behavioral differences are demonstrated and discussed. In addition, we compare these solutions to naturally occurring shallow water waves on beaches.

Towards A Science Of Human Stories: Using Sentiment Analysis And Emotional Arcs To Understand The Building Blocks Of Complex Social Systems, Andrew James Reagan

Graduate College Dissertations and Theses

We can leverage data and complex systems science to better understand society and human nature on a population scale through language --- utilizing tools that include sentiment analysis, machine learning, and data visualization.

Data-driven science and the sociotechnical systems that we use every day are enabling a transformation from hypothesis-driven, reductionist methodology to complex systems sciences.

Namely, the emergence and global adoption of social media has rendered possible the real-time estimation of population-scale sentiment, with profound implications for our understanding of human behavior.

Advances in computing power, natural language processing, and digitization of text now make it possible to study …

Hilbert Class Fields Of Imaginary Quadratic Fields And Reflex Fields Of Certain Sextic Cm Fields, Garvin Gaston

Graduate College Dissertations and Theses

In this thesis we look at particular details of class field theory for complex multiplication fields. We begin by giving some background on fields, abelian varieties, and complex multiplication. We then turn to the first task of this thesis and give an implementation in Sage of a classical algorithm to compute the Hilbert class field of a quadratic complex multiplication field using the j-invariant of elliptic curves with complex multiplication by the ring of integers of the field, and we include three explicit examples to illustrate the algorithm.

The second part of this thesis contains new results: Let K be …