Digital Commons Network^™

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 …

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.

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.

Higher-Order Nonstandard Finite Difference Methods For Autonomous Differential Equations With Applications In Mathematical Ecology, Fawaz Karhan R Alalhareth

Mathematics Dissertations - Archive

Nonstandard finite difference (NSFD) methods have been widely used to numerically solve various problems in biology. In recent years, NSFD methods have been proposed that preserve essential properties of the solutions of general autonomous differential equations, such as positivity and elementary stability, among others. However, those methods are only of first-order accuracy. In the first part of this dissertation, we construct and analyze two second-order modified positive and elementary stable nonstandard (PESN) numerical methods for n-dimensional autonomous differential equations. The new PESN methods are generalized versions of the explicit Euler's method and second-order accurate, thereby improving the order of accuracy …

Quantization For A Nonuniform Triadic Cantor Distribution, Asha Barua

Theses and Dissertations

The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. Let P be a Borel probability measure on R such that P := 1/4 P◦ S1−1 + 1\2 P◦ S2−1 + 1/4 P◦ S3−1, where S1, S2 and S3 are three contractive similarity mappings such that Sj(x) = 1/5x+2(j−1)/5, for all x ∈ R. For this probability measure, in this thesis, we determine the optimal sets of n-means and the nth quantization errors for …

Mathematics Teachers’ Working With Cooperative Learning, Jaime Gomez

Theses and Dissertations

Teaching styles vary greatly amongst educators. One being extensively researched and highly discussed is the method of cooperative learning. Although many studies have shown the benefits of incorporating cooperative learning into classrooms, it has not been a widely used method of teaching in high school mathematics classrooms. This study explores some of the efforts that teachers, who utilize cooperative learning in their classrooms, make to implement cooperative learning lessons successfully. Furthermore, this study also explores the challenges these teachers have encountered when using cooperative learning. Data was collected qualitatively by interviews and surveys from six in-service high …

Thermal Convection In A Cylindrical Annulus Filled With Porous Material, Anirban Ray

Theses and Dissertations

Here a study on thermal convection in a porous vertical cylindrical annulus which is heated from below is carried out. The walls are considered to be impermeable that is the velocity is 0 at the boundary walls. The cylindrical annulus is radially insulated. The governing system consists of the continuity equation, Darcy-Boussinesq equation, heat equation and the equation of state. Employing weakly non-linear approach, the basic state system and the perturbed system are derived. After obtaining the solutions to the basic state system, the pressure term in perturbed system is eliminated by taking double curl, and then eliminating the velocity, …

Verifying Sudoku Puzzles, Chelsea Schweer

Electronic Theses, Projects, and Dissertations

Sudoku puzzles, created by Meki Kaji around 1983, consist of a square 9 by 9 grid made up of 9 rows, 9 columns, and nine 3 by 3 square sub-grids called blocks. The goal of the puzzle is to be able to place the numbers 1 through 9 in every row, column, and block where no number is repeated in each row, column, and block. Imagine being given a completed Sudoku puzzle and having to check that it was solved correctly. You could just check all the rows columns and blocks (27 items), but is there a smaller number of …

Symmetric Generations And An Algorithm To Prove Relations, Diddier Andrade

Electronic Theses, Projects, and Dissertations

In this thesis we have discovered homomorphic images of several progenitors such as 3^(*56):(23:(3:7), 3^(*14):(23:(3:7)), 5^(∗24) : S5, 2^(∗10) : (10 : 2), 56^(∗24) : (A5 : 2), and 11^(∗12) :m L2(11). We give isomorphism types of each image that we have found.
We then create a monomial representation of L2(11) by lifting 5:11 onto it.
We manually perform Double Coset Enumeration of 3:(2×S5) over D12
to create its Cayley graph. This is achieved by solving many word problems. The
Cayley graph is used to find a permutation representation of 3:(2×S5). We also
perform Double Coset Enumeration S3 × A5 …

Development Of Graphical Models And Statistical Physics Motivated Approaches To Genomic Investigations, Yashwanth Lagisetty

Dissertations and Theses (Open Access)

Identifying genes involved in disease pathology has been a goal of genomic research since the early days of the field. However, as technology improves and the body of research grows, we are faced with more questions than answers. Among these is the pressing matter of our incomplete understanding of the genetic underpinnings of complex diseases. Many hypotheses offer explanations as to why direct and independent analyses of variants, as done in genome-wide association studies (GWAS), may not fully elucidate disease genetics. These range from pointing out flaws in statistical testing to invoking the complex dynamics of epigenetic processes. In the …

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 …

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 …

Interrogation Of Social Justice Contexts In Mathematical Modeling: The Use Of Simulations Of Practice In The Mathematical Preparation Of Teachers, Cynthia Oropesa Anhalt, Ricardo Cortez, Brynja Kohler, Will Tidwell

Mathematics and Statistics Faculty Publications

Research in prospective teachers’ development of mathematical modeling knowledge for teaching is gaining momentum. The Mathematics of Doing, Understanding, Learning, and Educating for Secondary Students [MODULE(S2)]* project developed a curriculum in modeling for teacher education that includes simulations of practice, in which prospective teachers reflect on and plan a discussion around student thinking, their models, and the contextualization of their results. We present an analysis of prospective teachers’ modeling work on the decreasing area of Indigenous reservation land in the U.S., and a simulation of practice which explores different methods for finding the area of land in connection to …

Using Teacher Noticing And Video-Mediated Professional Learning To Develop Preservice Teachers’ Knowledge For Teaching The Derivative, Alfred M. Limbere

Theses, Dissertations and Culminating Projects

This study investigated how problem-solving videos can be used in video-mediated professional learning to support secondary preservice mathematics teachers (PMTs) in developing teacher knowledge for noticing student thinking in the context of the derivative concept in calculus. A model of the trajectory of PMTs’ noticing was constructed as six PMTs viewed and analyzed videos of students’ problem solving. At the same time, the nature of video-mediated interactions that were found to be productive in supporting this knowledge development was examined. A design experiment was used as the research methodology. Data was collected from video recordings of eight semi-structured teaching episodes …

A Gpu Accelerated Genetic Algorithm For The Construction Of Hadamard Matrices, Andras Balogh, Raven Ruiz

School of Mathematical & Statistical Sciences Faculty Publications

We use a genetic algorithm to construct Hadamard Matrices. The initial population of random matrices is generated to have a balanced number of +1 and −1 entries in each column except the first column with all +1. Several fitness functions are implemented in order to find the most effective one. The crossover process creates offspring matrix population by exchanging columns of the parent matrix population. The mutation process flips +1 and −1 entry pairs in random columns. The use of CuPylibrary in Python on graphics processing units enables us to handle populations of thousands of matrices and matrix operations in …

Inexact Fixed-Point Proximity Algorithms For Nonsmooth Convex Optimization, Jin Ren

Mathematics & Statistics Theses & Dissertations

The aim of this dissertation is to develop efficient inexact fixed-point proximity algorithms with convergence guaranteed for nonsmooth convex optimization problems encountered in data science. Nonsmooth convex optimization is one of the core methodologies in data science to acquire knowledge from real-world data and has wide applications in various fields, including signal/image processing, machine learning and distributed computing. In particular, in the context of image reconstruction, compressed sensing and sparse machine learning, either the objective functions or the constraints of the modeling optimization problems are nondifferentiable. Hence, traditional methods such as the gradient descent method and the Newton method are …

Traveling Wave Solutions For The Negative Order Hierarchy Of The D-Akns Equations, Brayton Isaac Wario

Theses and Dissertations

In the thesis work, based on the D-AKNS spectral problem, we study the negative-order D-AKNS (ND-AKNS) hierarchy. In particular, the first ND-AKNS equation is derived from the negative-order D-AKNS hierarchy, which is proved integrable in the sense of Lax pair. Furthermore, we discuss the traveling wave solutions to the ND-AKNS Equation, including possible soliton solutions.

Bridge Trisections And Classical Knotted Surface Theory, Jason Joseph, Jeffrey Meier, Maggie Miller, Miller Zupan

Department of Mathematics: Faculty Publications

We seek to connect ideas in the theory of bridge trisections with other wellstudied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney–Massey theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of …

Controlling Textures In Tsp Art, Robert Bosch, Robert Klock

Faculty & Staff Scholarship

By making simple modifications to how we calculate distances between points/cities, we control textures in TSP Art.

Investigating Elementary School Students’ Reasoning About Dynamic Angles, Erell Germia

Theses, Dissertations and Culminating Projects

Angle measurement is a significant topic in almost all areas of mathematics learning and also in many disciplines outside mathematics education, such as engineering and architecture. According to the literature, there are three common conceptions of angles – as union of rays, rotations, and wedges. Researchers argued that students must consider these three angle concepts together to construct a meaningful understanding of angles. However, the curriculum standards for mathematics often present these angle conceptions separately to students, probably resulting in a fragmented understanding of the angle concept. In addition to this problem, the research literature documents multiple alternative conceptions that …

Weakly Q-Convex Domains And Bounded Q-Subharmonic Exhaustion Functions, Emily Foss

Graduate Theses and Dissertations

We generalize the Diederich-Fornaess index to bounded weakly q-convex domains with bounded q-subharmonic exhaustion functions. Sufficient conditions for this generalized Diederich-Fornæss index to have a given lower bound are proved. We show this generalized index is positive on bounded weakly q-convex domains with C^3 boundaries. Additionally, we prove sufficient conditions for this generalized index to equal one. For example, we show that if the domain has Property ( ̃(Pq ) ) then the domain has high hyperconvexity.

Improving Computation For Hierarchical Bayesian Spatial Gaussian Mixture Models With Application To The Analysis Of Thz Image Of Breast Tumor, Jean Remy Habimana

Graduate Theses and Dissertations

In the first chapter of this dissertation we give a brief introduction to Markov chain Monte Carlo methods (MCMC) and their application in Bayesian inference. In particular, we discuss the Metropolis-Hastings and conjugate Gibbs algorithms and explore the computational underpinnings of these methods. The second chapter discusses how to incorporate spatial autocorrelation in linear a regression model with an emphasis on the computational framework for estimating the spatial correlation patterns.

The third chapter starts with an overview of Gaussian mixture models (GMMs). However, because in the GMM framework the observations are assumed to be independent, GMMs are less effective when …

Machine Learning Model Comparison And Arma Simulation Of Exhaled Breath Signals Classifying Covid-19 Patients, Aaron Christopher Segura

Mathematics & Statistics ETDs

This study compared the performance of machine learning models in classifying COVID-19 patients using exhaled breath signals and simulated datasets. Ground truth classification was determined by the gold standard Polymerase Chain Reaction (PCR) test results. A residual bootstrapped method generated the simulated datasets by fitting signal data to Autoregressive Moving Average (ARMA) models. Classification models included neural networks, k-nearest neighbors, naïve Bayes, random forest, and support vector machines. A Recursive Feature Elimination (RFE) study was performed to determine if reducing signal features would improve the classification models performance using Gini Importance scoring for the two classes. The top 25% of …

Robust Uncertainty Quantification With Analysis Of Error In Standard And Non-Standard Quantities Of Interest, Zachary Stevens

Mathematics & Statistics ETDs

This thesis derives two Uncertainty Quantification (UQ) methods for differential equations that depend on random parameters: (\textbf{i}) error bounds for a computed cumulative distribution function (\textbf{ii}) a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria. Both UQ approaches utilize adjoint-based \textit{a posteriori} error analysis in order to accurately estimate the error in samples of numerically approximated quantities of interest. The adaptive MLMC algorithm developed in this thesis relies on the adjoint-based error analysis to adaptively create meshes and accurately monitor a stopping criteria. This is in contrast to classical MLMC algorithms which employ either a …

Visualizing Bivariate Statistics Using Ellipses Over A Scatter Plot, Mamunur Rashid, Jyotirmoy Sarkar, Siddhanta Phuyal

Mathematics Faculty Publications

A scatter plot shows the relationship between two quantitative variables 𝒙 and 𝒚. Sometimes, we can predict one variable as a linear function of the other using the least squares regression lines of 𝒚 on 𝒙 or 𝒙 on 𝒚. These two regression lines together suffice to identify the mean vector, the coefficient of determination, the correlation coefficient, and the ratio of the standard deviations (SD). So, do our proposed summary ellipses. Additionally, the inner ellipse reveals the SDs and the outer ellipse flags potential outliers.

Estimating The Parameters Of A Simple Linear Regression Model Without Using Differential Calculus, Mamunur Rashid, Jyotirmoy Sarkar

Mathematics Faculty Publications

To estimate the parameters of a simple linear regression model, students who already know calculus can minimize the total squared deviations by setting its first-order partial derivatives to zero and solving simultaneously. For students who do not know calculus, most teachers/textbooks simply state the formulas without justifying them. Students accept the formulas on faith; and for given data, they evaluate the estimates using a calculator or a statistical software. In this paper, we justify the formulas without invoking calculus. We hope the users of statistics will benefit from our proposed justifications.

Survivor Bond Models For Securitizing Longevity Risk, Priscilla Mansah Codjoe

Doctoral Dissertations

"Longevity risk is the risk that a reference population’s mortality rates deviate from what is projected from prior life tables. This is due to discoveries in biological sciences, improved public health measures, and nutrition, which have dramatically increased life expectancy. Longevity risk raises life insurers’ liability, increasing product costs and reserves. Securitization through longevity derivatives is a way of dealing with this risk.

To enhance the pricing of life contingent products, we present an additive type mortality model in the style of the Lee-Carter. This model incorporates policyholder covariates. By using counting processes and martingale machinery, we obtain close form …

Mathematical Modeling Of Seir Model With Generalized Incidence Function And The Extension To Covid-19 Model, Shymaa Mohammad Dadoa

Theses

The COVID-19 pandemic had shown the importance of the SEIR model in predicting the outcome of the disease spread and to find the best strategies to contain the pandemic. As this type of model has a limited number of compartments, many other models were derived from the SEIR model to cover, to the maximum, the complex dynamics of the disease spread. These extensions of the SEIR model bring natural validity questions: How can we validate these models? and how far/close are these extended models from giving us real insights into the pandemic?

This thesis investigates the SEIR epidemic model and …

Mathematical Constants Beyond The Half-Circle: An Open Call For Poetry, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

The Journal of Humanistic Mathematics invites submissions of mathematical poetry for an upcoming poetry folder tentatively titled Mathematical Constants Beyond the Half-Circle. Please send your submissions via email to the editors by November 1, 2022. Publication decisions will be made by December 31, 2022. The folder is expected to appear in January 2023.

Squate, Tom Blackford

Journal of Humanistic Mathematics

This is the story of a middle school student who befriends an irrational number, the square root of eight.