Physical Sciences and Mathematics Commons^™

Efficient Numerical Optimization For Parallel Dynamic Optimal Power Flow Simulation Using Network Geometry, Rylee Sundermann

Electronic Theses and Dissertations

In this work, we present a parallel method for accelerating the multi-period dynamic optimal power flow (DOPF). Our approach involves a distributed-memory parallelization of DOPF time-steps, use of a newly developed parallel primal-dual interior point method, and an iterative Krylov subspace linear solver with a block-Jacobi preconditioning scheme. The parallel primal-dual interior point method has been implemented and distributed in the open-source PETSc library and is currently available. We present the formulation of the DOPF problem, the developed primal dual interior point method solver, the parallel implementation, and results on various multi-core machines. We demonstrate the effectiveness our proposed block-Jacobi …

Investigaion Of The Gamma Hurdle Model For A Single Population Mean, Alissa Jacobs

Electronic Theses and Dissertations

A common issue in some statistical inference problems is dealing with a high frequency of zeroes in a sample of data. For many distributions such as the gamma, optimal inference procedures do not allow for zeroes to be present. In practice, however, it is natural to observe real data sets where nonnegative distributions would make sense to model but naturally zeroes will occur. One example of this is in the analysis of cost in insurance claim studies. One common approach to deal with the presence of zeroes is using a hurdle model. Most literary work on hurdle models will focus …

Totally Multicolored Rado Numbers For The Equation X_1 + X_2 + X_3 + ... + X_(M−1) = X_M, Skylar Halverson

Electronic Theses and Dissertations

A set is called Totally Multicolored (TMC) if no elements in the set are colored the same. For all natural numbers t, m, let R(t, m) be the least natural number n such that for every t-coloring of the set {1, 2, 3, ..., R(t, m)} there exist a solution set {x_1, x_2, . . ., x_m} to L(m), x_1 + x_2 + x_3 + ... + x_(m−1) = x_m such that x_i does not equal x_j for all i that does not equal j, that avoids being Totally Multicolored. This paper shows a function to find R(t,m) for any …

The Efficacy Of The South Dakota State University Summer Jacks Leap Program, Tessa M. Sundermann

Electronic Theses and Dissertations

Today, several studies detail the continuing struggle many students have with college mathematics courses at universities across the United States. The South Dakota State University Summer Jacks LeaP program is a summer bridge mathematics program aimed at improving incoming students’ mathematics success. This analysis used a mixed methods research design to examine the efficacy of the Summer Jacks LeaP program. First, we analyzed the LeaP students’ homework averages, exam 1 scores, final exam scores, and overall grade scores to determine if they were finding success in their fall semester mathematics courses. We also used hypothesis testing to compare LeaP participants …

Graph Realizability And Factor Properties Based On Degree Sequence, Daniel John

Electronic Theses and Dissertations

A graph is a structure consisting of a set of vertices and edges. Graph construction has been a focus of research for a long time, and generating graphs has proven helpful in complex networks and artificial intelligence.

A significant problem that has been a focus of research is whether a given sequence of integers is graphical. Havel and Hakimi stated necessary and sufficient conditions for a degree sequence to be graphic with different properties. In our work, we have proved the sufficiency of the requirements by generating algorithms and providing constructive proof.

Given a degree sequence, one crucial problem is …

The Equality Case Of The Kraft And The Kraft-Mcmillan Inequalities, Xavier Nunes

Electronic Theses and Dissertations

In this thesis, we analyze the Kraft Inequality and the Kraft-McMillan Inequality in their equality cases. Kraft’s Inequality deals with prefix-free code and Kraft-McMillan’s Inequality deals with uniquely decodable codes. The focus of the Kraft Inequality analysis is to study the occurrence of prefix-free codes that satisfy the equality case and the structure of words in the code when the equality condition is met. The second part of the thesis touches on the Kraft-McMillan Inequality. Since the proof of this latter inequality uses limits, we cannot immediately analyse its equality cases. The paper will therefore study the equality cases of …

Cryptography Through The Lens Of Group Theory, Dawson M. Shores

Electronic Theses and Dissertations

Cryptography has been around for many years, and mathematics has been around even longer. When the two subjects were combined, however, both the improvements and attacks on cryptography were prevalent. This paper introduces and performs a comparative analysis of two versions of the ElGamal cryptosystem, both of which use the specific field of mathematics known as group theory.

Reinforcement Learning: Low Discrepancy Action Selection For Continuous States And Actions, Jedidiah Lindborg

Electronic Theses and Dissertations

In reinforcement learning the process of selecting an action during the exploration or exploitation stage is difficult to optimize. The purpose of this thesis is to create an action selection process for an agent by employing a low discrepancy action selection (LDAS) method. This should allow the agent to quickly determine the utility of its actions by prioritizing actions that are dissimilar to ones that it has already picked. In this way the learning process should be faster for the agent and result in more optimal policies.

A Practical Extension To The Ab/Ba Design, My T.A Nguyen

Electronic Theses and Dissertations

In this work, we take a close look at a general extension to the traditional AB/BA
crossover design that is commonly used in clinical trials to determine the effectiveness
of new candidate drugs. While the traditional crossover design requires each patient
in the study to be measured on both treatment A and treatment B, we consider the
possibility of additional measurements being available on each patient. This produces
designs such as the AABB/BBAA design which has been used in previous studies.
A general test statistic will be derived to test for treatment effects as well as its
corresponding power function …

Decisive Neutrality, Restricted Decisive Neutrality, And Split Decisive Neutrality On Median Semilattices And Median Graphs., Ulf Högnäs

Electronic Theses and Dissertations

Consensus functions on finite median semilattices and finite median graphs are studied from an axiomatic point of view. We start with a new axiomatic characterization of majority rule on a large class of median semilattices we call sufficient. A key axiom in this result is the restricted decisive neutrality condition. This condition is a restricted version of the more well-known axiom of decisive neutrality given in [4]. Our theorem is an extension of the main result given in [7]. Another main result is a complete characterization of the class of consensus on a finite median semilattice that satisfies the axioms …

Partially Oriented 6-Star Decomposition Of Some Complete Mixed Graphs, Kazeem A. Kosebinu

Electronic Theses and Dissertations

Let $M_v$ denotes a complete mixed graph on $v$ vertices, and let $S_6^i$ denotes the partial orientation of the 6-star with twice as many arcs as edges. In this work, we state and prove the necessary and sufficient conditions for the existence of $\lambda$-fold decomposition of a complete mixed graph into $S_6^i$ for $i\in\{1,2,3,4\}$. We used the difference method for our proof in some cases. We also give some general sufficient conditions for the existence of $S_6^i$-decomposition of the complete bipartite mixed graph for $i\in\{1,2,3,4\}$. Finally, this work introduces the decomposition of a complete mixed graph with a hole into …

Applying Deep Learning To The Ice Cream Vendor Problem: An Extension Of The Newsvendor Problem, Gaffar Solihu

Electronic Theses and Dissertations

The Newsvendor problem is a classical supply chain problem used to develop strategies for inventory optimization. The goal of the newsvendor problem is to predict the optimal order quantity of a product to meet an uncertain demand in the future, given that the demand distribution itself is known. The Ice Cream Vendor Problem extends the classical newsvendor problem to an uncertain demand with unknown distribution, albeit a distribution that is known to depend on exogenous features. The goal is thus to estimate the order quantity that minimizes the total cost when demand does not follow any known statistical distribution. The …

Manifold Learning With Tensorial Network Laplacians, Scott Sanders

Electronic Theses and Dissertations

The interdisciplinary field of machine learning studies algorithms in which functionality is dependent on data sets. This data is often treated as a matrix, and a variety of mathematical methods have been developed to glean information from this data structure such as matrix decomposition. The Laplacian matrix, for example, is commonly used to reconstruct networks, and the eigenpairs of this matrix are used in matrix decomposition. Moreover, concepts such as SVD matrix factorization are closely connected to manifold learning, a subfield of machine learning that assumes the observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. Since …

Multilateration Index., Chip Lynch

Electronic Theses and Dissertations

We present an alternative method for pre-processing and storing point data, particularly for Geospatial points, by storing multilateration distances to fixed points rather than coordinates such as Latitude and Longitude. We explore the use of this data to improve query performance for some distance related queries such as nearest neighbor and query-within-radius (i.e. “find all points in a set P within distance d of query point q”). Further, we discuss the problem of “Network Adequacy” common to medical and communications businesses, to analyze questions such as “are at least 90% of patients living within 50 miles of a covered emergency …

Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

Electronic Theses and Dissertations

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …

Constructions & Optimization In Classical Real Analysis Theorems, Abderrahim Elallam

Electronic Theses and Dissertations

This thesis takes a closer look at three fundamental Classical Theorems in Real Analysis. First, for the Bolzano Weierstrass Theorem, we will be interested in constructing a convergent subsequence from a non-convergent bounded sequence. Such a subsequence is guaranteed to exist, but it is often not obvious what it is, e.g., if an = sin n. Next, the H¨older Inequality gives an upper bound, in terms of p ∈ [1,∞], for the the integral of the product of two functions. We will find the value of p that gives the best (smallest) upper-bound, focusing on the Beta and Gamma integrals. …

Comparison Of Software Packages For Detecting Differentially Expressed Genes From Single-Sample Rna-Seq Data, Rong Zhou

Electronic Theses and Dissertations

RNA-sequencing (RNA-seq) has rapidly become the tool in many genome-wide transcriptomic studies. It provides a way to understand the RNA environment of cells in different physiological or pathological states to determine how cells respond to these changes. RNA-seq provides quantitative information about the abundance of different RNA species present in a given sample. If the difference or change observed in the read counts or expression level between two experimental conditions is statistically significant, the gene is declared as differentially expressed. A large number of methods for detecting differentially expressed genes (DEGs) with RNA-seq have been developed, such as the methods …

Zn Orbifolds Of Vertex Operator Algebras, Daniel Graybill

Electronic Theses and Dissertations

Given a vertex algebra V and a group of automorphisms of V, the invariant subalgebra V^G is called an orbifold of V. This construction appeared first in physics and was also fundamental to the construction of the Moonshine module in the work of Borcherds. It is expected that nice properties of V such as C2-cofiniteness and rationality will be inherited by V^G if G is a finite group. It is also expected that under reasonable hypotheses, if V is strongly finitely generated and G is reductive, V^G will also be strongly finitely generated. This is an analogue …

Exponential Random Graphs And A Generalization Of Parking Functions, Ryan Demuse

Electronic Theses and Dissertations

Random graphs are a powerful tool in the analysis of modern networks. Exponential random graph models provide a framework that allows one to encode desirable subgraph features directly into the probability measure. Using the theory of graph limits pioneered by Borgs et. al. as a foundation, we build upon the work of Chatterjee & Diaconis and Radin & Yin. We add complexity to the previously studied models by considering exponential random graph models with edge-weights coming from a generic distribution satisfying mild assumptions. In particular, we show that a large family of two-parameter, edge-weighted exponential random graphs display a phase …

Detailing The Connection Between A Family Of Polar Graphs And Tremain Equiangular Tight Frames, Nicholas Brown

Electronic Theses and Dissertations

The relationship between strongly regular graphs and equiangular tight frames has been known for several years, and this relationship has been used to construct many of the most recent examples of new strongly regular graphs. In this paper, we present an explicit construction of a family of equiangular tight frames using the geometry of a quadratic space over the field of four elements. We observe that these frames give rise to a strongly regular graph on a subset of points of a quadratic space over the field with 4 elements. We then demonstrate an isomorphism between this graph and a …

Topics On Applications Of Optimization Theories On Statistical Methodologies, Duc Anh Anh Doan

Electronic Theses and Dissertations

In this dissertation, We show the results of our researches in statistical sampling, functional optimization, and methodology for partially observed Markov process (POMP) models. In statistical sampling, we introduce a p-generalized smoothing method that enables the Langevin-Monte Carlo method to generate a sample from a log concave distribution weakly smoothing potential function. For our optimization research, we introduce an accelerated inexact gradient (AIG) method. Combining the strengths while mitigating the weakness of its parent methods: gradient descent and Nesterov's accelerated gradient, AIG converges with excellent rates for both convex and non-convex optimization problems for smooth objective functions. Furthermore, we also …

On Domination And Bondage Numbers Of Some Classes Of Graphs, Andrew Pham

Electronic Theses and Dissertations

Given a simple finite graph G=(V,E), a vertex subset D ? V(G) is said to be a dominating set of G if every vertex v ? V(G)-D is adjacent to a vertex in D. The domination number of G, denoted ?(G), is the minimum cardinality among all dominating sets of G. In a network, the domination number determines the minimum number of sites required to dominate the entire network at a minimum cost. The bondage number of a graph G is the minimum cardinality among all edge sets B such that ?(G-B) > ?(G). The bondage number may serve as a …

Numerical Approximation Of Lyapunov Exponents And Its Applications In Control Systems, Nakita K. Andrews

Electronic Theses and Dissertations

The progression of state trajectories with respect to time, and its stability properties can be described by a system of nonlinear differential equations. However, since most nonlinear dynamical systems cannot be solved by hand, one must rely on computer simulations to observe the behavior of the system. This work focuses on chaotic systems. The Lyapunov Exponent (LE) is frequently used in the quantitative studies of a chaotic system. Lyapunov exponents give the average rate of separation of nearby orbits in phase space, which can be used to determine the state of a system, e.g. stable or unstable. The objective of …

Decompositions Of The Complete Mixed Graph By Mixed Stars, Chance Culver

Electronic Theses and Dissertations

In the study of mixed graphs, a common question is: What are the necessary and suffcient conditions for the existence of a decomposition of the complete mixed graph into isomorphic copies of a given mixed graph? Since the complete mixed graph has twice as many arcs as edges, then an obvious necessary condition is that the isomorphic copies have twice as many arcs as edges. We will prove necessary and suffcient conditions for the existence of a decomposition of the complete mixed graphs into mixed stars with two edges and four arcs. We also consider some special cases of decompositions …

Calculating Infinite Series Using Parseval's Identity, James R. Poulin

Electronic Theses and Dissertations

Parseval's identity is an equality from Fourier analysis that relates an infinite series over the integers to an integral over an interval, which can be used to evaluate the exact value of some classes of infinite series. We compute the exact value of the Riemann zeta function at the positive even integers using the identity, and then we use it to compute the exact value of an infinite series whose summand is a rational function summable over the integers.

Gray Codes In Music Theory, Isaac L. Vaccaro

Electronic Theses and Dissertations

In the branch of Western music theory called serialism, it is desirable to construct chord progressions that use each chord in a chosen set exactly once. We view this problem through the scope of the mathematical theory of Gray codes, the notion of ordering a finite set X so that adjacent elements are related by an element of some specified set R of involutions in the permutation group of X. Using some basic results from the theory of permutation groups we translate the problem of finding Gray codes into the problem of finding Hamiltonian paths and cycles in a Schreier …

Trees With Unique Italian Dominating Functions Of Minimum Weight, Alyssa England

Electronic Theses and Dissertations

An Italian dominating function, abbreviated IDF, of $G$ is a function $f \colon V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that for every vertex $v \in V(G)$ with $f(v)=0$, we have $\sum_{u \in N(v)} f(u) \ge 2$. That is, either $v$ is adjacent to at least one vertex $u$ with $f(u) = 2$, or to at least two vertices $x$ and $y$ with $f(x) = f(y) = 1$. The Italian domination number, denoted $\gamma_I$(G), is the minimum weight of an IDF in $G$. In this thesis, we use operations that join two trees with a single edge in order …

An Analysis Of The First Passage To The Origin (Fpo) Distribution, Aradhana Soni

Electronic Theses and Dissertations

What is the probability that in a fair coin toss game (a simple random walk) we go bankrupt in n steps when there is an initial lead of some known or unknown quantity $m? What is the distribution of the number of steps N that it takes for the lead to vanish? This thesis explores some of the features of this first passage to the origin (FPO) distribution. First, we explore the distribution of N when m is known. Next, we compute the maximum likelihood estimators of m for a fixed n and also the posterior distribution of m when …

Novel Inference Methods For Generalized Linear Models Using Shrinkage Priors And Data Augmentation., Arinjita Bhattacharyya

Electronic Theses and Dissertations

Generalized linear models have broad applications in biostatistics and sociology. In a regression setup, the main target is to find a relevant set of predictors out of a large collection of covariates. Sparsity is the assumption that only a few of these covariates in a regression setup have a meaningful correlation with an outcome variate of interest. Sparsity is incorporated by regularizing the irrelevant slopes towards zero without changing the relevant predictors and keeping the resulting inferences intact. Frequentist variable selection and sparsity are addressed by popular techniques like Lasso, Elastic Net. Bayesian penalized regression can tackle the curse of …

Discrepancy Inequalities In Graphs And Their Applications, Adam Purcilly

Electronic Theses and Dissertations

Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based …