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Convolution And Autoencoders Applied To Nonlinear Differential Equations, Noah Borquaye
Convolution And Autoencoders Applied To Nonlinear Differential Equations, Noah Borquaye
Electronic Theses and Dissertations
Autoencoders, a type of artificial neural network, have gained recognition by researchers in various fields, especially machine learning due to their vast applications in data representations from inputs. Recently researchers have explored the possibility to extend the application of autoencoders to solve nonlinear differential equations. Algorithms and methods employed in an autoencoder framework include sparse identification of nonlinear dynamics (SINDy), dynamic mode decomposition (DMD), Koopman operator theory and singular value decomposition (SVD). These approaches use matrix multiplication to represent linear transformation. However, machine learning algorithms often use convolution to represent linear transformations. In our work, we modify these approaches to …
A Bridge Between Graph Neural Networks And Transformers: Positional Encodings As Node Embeddings, Bright Kwaku Manu
A Bridge Between Graph Neural Networks And Transformers: Positional Encodings As Node Embeddings, Bright Kwaku Manu
Electronic Theses and Dissertations
Graph Neural Networks and Transformers are very powerful frameworks for learning machine learning tasks. While they were evolved separately in diverse fields, current research has revealed some similarities and links between them. This work focuses on bridging the gap between GNNs and Transformers by offering a uniform framework that highlights their similarities and distinctions. We perform positional encodings and identify key properties that make the positional encodings node embeddings. We found that the properties of expressiveness, efficiency and interpretability were achieved in the process. We saw that it is possible to use positional encodings as node embeddings, which can be …
Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun
Roots Of Quaternionic Polynomials And Automorphisms Of Roots, Olalekan Ogunmefun
Electronic Theses and Dissertations
The quaternions are an extension of the complex numbers which were first described by Sir William Rowan Hamilton in 1843. In his description, he gave the equation of the multiplication of the imaginary component similar to that of complex numbers. Many mathematicians have studied the zeros of quaternionic polynomials. Prominent of these, Ivan Niven pioneered a root-finding algorithm in 1941, Gentili and Struppa proved the Fundamental Theorem of Algebra (FTA) for quaternions in 2007. This thesis finds the zeros of quaternionic polynomials using the Fundamental Theorem of Algebra. There are isolated zeros and spheres of zeros. In this thesis, we …
Enestr¨Om-Kakeya Type Results For Complex And Quaternionic Polynomials, Matthew Gladin
Enestr¨Om-Kakeya Type Results For Complex And Quaternionic Polynomials, Matthew Gladin
Electronic Theses and Dissertations
The well known Eneström-Kakeya Theorem states that: for P(z)=∑i=0n ai zi, a polynomial of degree n with real coefficients satisfying 0 ≤ a0 ≤ a1 ≤ ⋯≤ an, all zeros of P(z) lie in |z|≤1 in the complex plane. In this thesis, we will find inner and outer bounds in which the zeros of complex and quaternionic polynomials lie. We will do this by imposing restrictions on the real and imaginary parts, and on the moduli, of the complex and quaternionic coefficients. We also apply similar restrictions on complex polynomials with …
John Horton Conway: The Man And His Knot Theory, Dillon Ketron
John Horton Conway: The Man And His Knot Theory, Dillon Ketron
Electronic Theses and Dissertations
John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.
Partially Oriented 6-Star Decomposition Of Some Complete Mixed Graphs, Kazeem A. Kosebinu
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
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
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 …
Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang
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
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. …
Decompositions Of The Complete Mixed Graph By Mixed Stars, Chance Culver
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 …
Trees With Unique Italian Dominating Functions Of Minimum Weight, Alyssa England
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
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 …
Hybrid Recommender Systems Via Spectral Learning And A Random Forest, Alyssa Williams
Hybrid Recommender Systems Via Spectral Learning And A Random Forest, Alyssa Williams
Electronic Theses and Dissertations
We demonstrate spectral learning can be combined with a random forest classifier to produce a hybrid recommender system capable of incorporating meta information. Spectral learning is supervised learning in which data is in the form of one or more networks. Responses are predicted from features obtained from the eigenvector decomposition of matrix representations of the networks. Spectral learning is based on the highest weight eigenvectors of natural Markov chain representations. A random forest is an ensemble technique for supervised learning whose internal predictive model can be interpreted as a nearest neighbor network. A hybrid recommender can be constructed by first …
Period Estimation And Denoising Families Of Nonuniformly Sampled Time Series, William Seguine
Period Estimation And Denoising Families Of Nonuniformly Sampled Time Series, William Seguine
Electronic Theses and Dissertations
Nonuniformly sampled time series are common in astronomy, finance, and other areas of research. Commonly, these time series belong to a family of signals recorded from the same phenomenon. Period estimation and denoising of such data relies on periodograms. In particular, the Lomb-Scargle periodogram and its extension, the Multiband Lomb-Scargle, are at the forefront of time series period estimation. However, these methods are not without laws. This paper explores alternatives to the Lomb-Scargle and Multiband Lomb-Scargle. In particular, this thesis uses regularized least squares and the convolution theorem to introduce a spectral consensus model of a family of nonuniformly sampled …
Roman Domination Cover Rubbling, Nicholas Carney
Roman Domination Cover Rubbling, Nicholas Carney
Electronic Theses and Dissertations
In this thesis, we introduce Roman domination cover rubbling as an extension of domination cover rubbling. We define a parameter on a graph $G$ called the \textit{Roman domination cover rubbling number}, denoted $\rho_{R}(G)$, as the smallest number of pebbles, so that from any initial configuration of those pebbles on $G$, it is possible to obtain a configuration which is Roman dominating after some sequence of pebbling and rubbling moves. We begin by characterizing graphs $G$ having small $\rho_{R}(G)$ value. Among other things, we also obtain the Roman domination cover rubbling number for paths and give an upper bound for the …
Generalizations Of The Arcsine Distribution, Rebecca Rasnick
Generalizations Of The Arcsine Distribution, Rebecca Rasnick
Electronic Theses and Dissertations
The arcsine distribution looks at the fraction of time one player is winning in a fair coin toss game and has been studied for over a hundred years. There has been little further work on how the distribution changes when the coin tosses are not fair or when a player has already won the initial coin tosses or, equivalently, starts with a lead. This thesis will first cover a proof of the arcsine distribution. Then, we explore how the distribution changes when the coin the is unfair. Finally, we will explore the distribution when one person has won the first …
Taking Notes: Generating Twelve-Tone Music With Mathematics, Nathan Molder
Taking Notes: Generating Twelve-Tone Music With Mathematics, Nathan Molder
Electronic Theses and Dissertations
There has often been a connection between music and mathematics. The world of musical composition is full of combinations of orderings of different musical notes, each of which has different sound quality, length, and em phasis. One of the more intricate composition styles is twelve-tone music, where twelve unique notes (up to octave isomorphism) must be used before they can be repeated. In this thesis, we aim to show multiple ways in which mathematics can be used directly to compose twelve-tone musical scores.
Perfect Double Roman Domination Of Trees, Ayotunde Egunjobi
Perfect Double Roman Domination Of Trees, Ayotunde Egunjobi
Electronic Theses and Dissertations
See supplemental content for abstract
Italian Domination On Ladders And Related Products, Bradley Gardner
Italian Domination On Ladders And Related Products, Bradley Gardner
Electronic Theses and Dissertations
An Italian dominating function on a graph $G = (V,E)$ is a function such that $f : V \to \{0,1,2\}$, and for each vertex $v \in V$ for which $f(v) = 0$, we have $\sum_{u\in N(v)}f(u) \geq 2$. The weight of an Italian dominating function is $f(V) = \sum_{v\in V(G)}f(v)$. The minimum weight of all such functions on a graph $G$ is called the Italian domination number of $G$. In this thesis, we will consider Italian domination in various types of products of a graph $G$ with the complete graph $K_2$. We will find the value of the Italian domination …
The Expected Number Of Patterns In A Random Generated Permutation On [N] = {1,2,...,N}, Evelyn Fokuoh
The Expected Number Of Patterns In A Random Generated Permutation On [N] = {1,2,...,N}, Evelyn Fokuoh
Electronic Theses and Dissertations
Previous work by Flaxman (2004) and Biers-Ariel et al. (2018) focused on the number of distinct words embedded in a string of words of length n. In this thesis, we will extend this work to permutations, focusing on the maximum number of distinct permutations contained in a permutation on [n] = {1,2,...,n} and on the expected number of distinct permutations contained in a random permutation on [n]. We further considered the problem where repetition of subsequences are as a result of the occurrence of (Type A and/or Type B) replications. Our method of enumerating the Type A replications causes double …
The 2-Domination Number Of A Caterpillar, Presley Chukwukere
The 2-Domination Number Of A Caterpillar, Presley Chukwukere
Electronic Theses and Dissertations
A set D of vertices in a graph G is a 2-dominating set of G if every vertex in V − D has at least two neighbors in D. The 2-domination number of a graph G, denoted by γ2(G), is the minimum cardinality of a 2- dominating set of G. In this thesis, we discuss the 2-domination number of a special family of trees, called caterpillars. A caterpillar is a graph denoted by Pk(x1, x2, ..., xk), where xi is the number of leaves attached to the ith vertex …
Developing Optimization Techniques For Logistical Tendering Using Reverse Combinatorial Auctions, Jennifer Kiser
Developing Optimization Techniques For Logistical Tendering Using Reverse Combinatorial Auctions, Jennifer Kiser
Electronic Theses and Dissertations
In business-to-business logistical sourcing events, companies regularly use a bidding process known as tendering in the procurement of transportation services from third-party providers. Usually in the form of an auction involving a single buyer and one or more sellers, the buyer must make decisions regarding with which suppliers to partner and how to distribute the transportation lanes and volume among its suppliers; this is equivalent to solving the optimization problem commonly referred to as the Winner Determination Problem. In order to take into account the complexities inherent to the procurement problem, such as considering a supplier’s network, economies of scope, …
Vector Partitions, Jennifer French
Vector Partitions, Jennifer French
Electronic Theses and Dissertations
Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose …
A Study Of Topological Invariants In The Braid Group B2, Andrew Sweeney
A Study Of Topological Invariants In The Braid Group B2, Andrew Sweeney
Electronic Theses and Dissertations
The Jones polynomial is a special topological invariant in the field of Knot Theory. Created by Vaughn Jones, in the year 1984, it is used to study when links in space are topologically different and when they are topologically equivalent. This thesis discusses the Jones polynomial in depth as well as determines a general form for the closure of any braid in the braid group B2 where the closure is a knot. This derivation is facilitated by the help of the Temperley-Lieb algebra as well as with tools from the field of Abstract Algebra. In general, the Artin braid group …
Italian Domination In Complementary Prisms, Haley D. Russell
Italian Domination In Complementary Prisms, Haley D. Russell
Electronic Theses and Dissertations
Let $G$ be any graph and let $\overline{G}$ be its complement. The complementary prism of $G$ is formed from the disjoint union of a graph $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. An Italian dominating function on a graph $G$ is a function such that $f \, : \, V \to \{ 0,1,2 \}$ and for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N(v)} f(u) \geq 2$. The weight of an Italian dominating function is the value $f(V)=\sum_{u \in V(G)}f(u)$. …
Vertex Weighted Spectral Clustering, Mohammad Masum
Vertex Weighted Spectral Clustering, Mohammad Masum
Electronic Theses and Dissertations
Spectral clustering is often used to partition a data set into a specified number of clusters. Both the unweighted and the vertex-weighted approaches use eigenvectors of the Laplacian matrix of a graph. Our focus is on using vertex-weighted methods to refine clustering of observations. An eigenvector corresponding with the second smallest eigenvalue of the Laplacian matrix of a graph is called a Fiedler vector. Coefficients of a Fiedler vector are used to partition vertices of a given graph into two clusters. A vertex of a graph is classified as unassociated if the Fiedler coefficient of the vertex is close to …
Application Of Symplectic Integration On A Dynamical System, William Frazier
Application Of Symplectic Integration On A Dynamical System, William Frazier
Electronic Theses and Dissertations
Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic …
On T-Restricted Optimal Rubbling Of Graphs, Kyle Murphy
On T-Restricted Optimal Rubbling Of Graphs, Kyle Murphy
Electronic Theses and Dissertations
For a graph G = (V;E), a pebble distribution is defined as a mapping of the vertex set in to the integers, where each vertex begins with f(v) pebbles. A pebbling move takes two pebbles from some vertex adjacent to v and places one pebble on v. A rubbling move takes one pebble from each of two vertices that are adjacent to v and places one pebble on v. A vertex x is reachable under a pebbling distribution f if there exists some sequence of rubbling and pebbling moves that places a pebble on x. A pebbling distribution where every …
Roman Domination In Complementary Prisms, Alawi I. Alhashim
Roman Domination In Complementary Prisms, Alawi I. Alhashim
Electronic Theses and Dissertations
The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. …