Mathematics Commons^™

Reliability Improvement On Feasibility Study For Selection Of Infrastructure Projects Using Data Mining And Machine Learning, Xi Hu

Theses and Dissertations

With the progressive development of infrastructure construction, conventional analytical methods such as correlation index, quantifying factors, and peer review are no longer satisfactory in support for decision-making of implementing an infrastructure project in the age of big data. This study proposes using a mathematical model named Fuzzy-Neural Comprehensive Evaluation Model (FNCEM) to improve the reliability of the feasibility study of infrastructure projects by using data mining and machine learning. Specifically, the data collection on time-series data, including traffic videos (278 Gigabytes) and historical weather data, uses transportation cameras and online searching, respectively. Meanwhile, the researcher sent out a questionnaire for …

Diameter Of 3-Colorable Graphs And Some Remarks On The Midrange Crossing Constant, Inne Singgih

Theses and Dissertations

The first part of this dissertation discussing the problem of bounding the diameter of a graph in terms of its order and minimum degree. The initial problem was solved independently by several authors between 1965 − 1989. They proved that for fixed δ ≥ 2 and large n, diam(G) ≤ 3n+ O(1). In 1989, Erdős, Pach, Pollack, and Tuza conjectured that the upper bound on the diameter can be improved if G does not contain a large complete subgraph K_k.

Let r, δ ≥ 2 be fixed integers and let G be a connected graph with n vertices …

Counting Number Fields By Discriminant, Harsh Mehta

Theses and Dissertations

The central topic of this dissertation is counting number fields ordered by discriminant. We fix a base field k and let Nd(k,G;X) be the number of extensions N/k up to isomorphism with Nk/Q(dN/k) ≤ X, [N : k] = d and the Galois closure of N/k is equal to G.

We establish two main results in this work. In the first result we establish upper bounds for N_|G| (k,G;X) in the case that G is a finite group with an abelian normal subgroup. Further, we establish upper bounds for the case N _|F| (k,G;X) where G is a Frobenius …

Rationality Questions And The Derived Category, Alicia Lamarche

Theses and Dissertations

This document is roughly divided into four chapters. The first outlines basic preliminary material, definitions, and foundational theorems required throughout the text. The second chapter, which is joint work with Dr. Matthew Ballard, gives an example of a family of Fano arithmetic toric varieties in which the derived category is able to detect the existence of k-rational points. More succinctly, we show that if X is a generalized del Pezzo variety defined over a field k, then X contains a k-rational point (and is in fact k-rational, that is, birational to Pnk ) if and only if Db(X) admits a …

Connections Between Extremal Combinatorics, Probabilistic Methods, Ricci Curvature Of Graphs, And Linear Algebra, Zhiyu Wang

Theses and Dissertations

This thesis studies some problems in extremal and probabilistic combinatorics, Ricci curvature of graphs, spectral hypergraph theory and the interplay between these areas. The first main focus of this thesis is to investigate several Ramsey-type problems on graphs, hypergraphs and sequences using probabilistic, combinatorial, algorithmic and spectral techniques:

• The size-Ramsey number Rˆ(G, r) is defined as the minimum number of edges in a hypergraph H such that every r-edge-coloring of H contains a monochromatic copy of G in H. We improved a result of Dudek, La Fleur, Mubayi and Rödl [ J. Graph Theory 2017 ] on the size-Ramsey number …

An Ensemble-Based Projection Method And Its Numerical Investigation, Shuai Yuan

Theses and Dissertations

In many cases, partial differential equation (PDE) models involve a set of parameters whose values may vary over a wide range in application problems, such as optimization, control and uncertainty quantification. Performing multiple numerical simulations in large-scale settings often leads to tremendous demands on computational resources. Thus, the ensemble method has been developed for accelerating a sequence of numerical simulations. In this work we first consider numerical solutions of Navier-Stokes equations under different conditions and introduce the ensemblebased projection method to reduce the computational cost. In particular, we incorporate a sparse grad-div stabilization into the method as a nonzero penalty …

Finite Axiomatisability In Nilpotent Varieties, Joshua Thomas Grice

Theses and Dissertations

Study of general algebraic systems has long been concerned with finite basis results that prove finite axiomatisability of certain classes of general algebras. In the 1970’s, Bjarni Jónsson speculated that a variety generated by a finite algebra might be finitely based provided the variety has a finite residual bound (that is, a finite bound on the cardinality of subdirectly irreducible algebras in the variety). As such, most finite basis results since then have had the hypothesis of a finite residual bound. However, Jónsson also speculated that it might be sufficient to replace the finite residual bound with the weaker hypothesis …

Distance Related Graph Invariants In Triangulations And Quadrangulations Of The Sphere, Trevor Vincent Olsen

Theses and Dissertations

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. I provide asymptotic upper bounds and sharp lower bounds for the Wiener index of simple triangulations and quadrangulations with given connectivity. Additionally, I make conjectures for the extremal triangulations and quadrangulations which maximize the Wiener index based on computational evidence. If σ(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness and proximity of G are defined as the largest and smallest value of σ(v) over all vertices v of G, respectively. …

Windows And Generalized Drinfeld Kernels, Robert R. Vandermolen

Theses and Dissertations

We develop a generalization of a construction of Drinfeld, first inspired by the Qconstruction of Ballard, Diemer, and Favero. We use this construction to provide kernels for Grassmann flops over an arbitrary field of characteristic zero. In the case of Grassmann flops this generalization recovers the kernel for a Fourier-Mukai functor on the derived category of the associated global quotient stack studied by Buchweitz, Leuschke, and Van den Bergh. We show an idempotent property for this kernel, which after restriction, induces a derived equivalence over any twisted form of a Grassmann flop.

Two Inquiries Related To The Digits Of Prime Numbers, Jeremiah T. Southwick

Theses and Dissertations

This dissertation considers two different topics. In the first part of the dissertation, we show that a positive proportion of the primes have the property that if any one of their digits in base 10, including their infinitely many leading 0 digits, is replaced by a different digit, then the resulting number is composite. We show that the same result holds for bases b 2 {2, 3, · · · , 8, 9, 11, 31}.

In the second part of the dissertation, we show for an integer b ≥ 5 that if a polynomial ƒ( x) with non-negative coefficients …

Congruences For Coefficients Of Modular Functions In Levels 3, 5, And 7 With Poles At 0, Ryan Austin Keck

Theses and Dissertations

We give congruences modulo powers of p in {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and Jenkins and continuing work done by the Jenkins, the author, and Moss. The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at infinity.

Symmetry Algebras Of The Canonical Lie Group Geodesic Equations In Dimension Five, Hassan Almusawa

Theses and Dissertations

Nowadays, there is much interest in constructing exact analytical solutions of differential equations using Lie symmetry methods. Lie devised the method in the 1880s. These methods were substantially developed utilizing modern mathematical language in the 1960s and 1970s by several different groups of authors such as L.V. Ovsiannikov, G. Bluman, and P. J. Olver, and have since been implemented as a software package for symbolic computation on commonly used platforms such as Mathematica and MAPLE.

In this work, we first develop an algorithmic scheme using the MAPLE platform to perform a Lie symmetry algebra identification and validate it on nonlinear …

Invariance And Invertibility In Deep Neural Networks, Han Zhang

Theses and Dissertations

Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input …

Constructing Associative Rings From Certain Hypergroups, Oscar Gonzalez

Theses and Dissertations

Tight hypergroups give rise to associative rings if the so-called general normality condition holds\cite{ARTICLE:1}. We consider four examples of tight hypergroups which do not satisfy the general normality condition and show that they still give rise to associative rings. Our examples are HM332(24),HM10353(32), HM10933(32) and HM10941(32)[1].

Moving Off Collections And Their Applications, In Particular To Function Spaces, Aaron Fowlkes

Theses and Dissertations

The main focus of this paper is the concept of a moving off collection of sets. We will be looking at how this relatively lesser known idea connects and interacts with other more widely used topological properties. In particular we will examine how moving off collections act with the function spaces Cp(X), C0(X), and CK (X). We conclude with a chapter on the Cantor tree and its moving off connections.

Many of the discussions of important theorems in the literature are expressed in terms that do not suggest the concept …

Numerical Methods For A Class Of Reaction-Diffusion Equations With Free Boundaries, Shuang Liu

Theses and Dissertations

The spreading behavior of new or invasive species is a central topic in ecology. The modelings of free boundary problems are widely studied to better understand the nature of spreading behavior of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of free boundary problems, due to the moving boundary, the stiffness of the system and topological changes.

In this work, we design numerical methods to investigate the spreading behavior of new species for a diffusive logistic model with a free boundary and a diffusive competition system with free boundaries. We develop a …

The Application Of Synthetic Signals For Ecg Beat Classification, Elliot Morgan Brown

Theses and Dissertations

A brief overview of electrocardiogram (ECG) properties and the characteristics of various cardiac conditions is given. Two different models are used to generate synthetic ECG signals. Domain knowledge is used to create synthetic examples of 16 different heart beat types with these models. Other techniques for synthesizing ECG signals are explored. Various machine learning models with different combinations of real and synthetic data are used to classify individual heart beats. The performance of the different methods and models are compared, and synthetic data is shown to be useful in beat classification.

2-Hyponormality On Unilateral Weighted Shifts, Juan G. Benitez

Theses and Dissertations

Given the concept of a normal operator, several weaker notions have been proposed in order to extend the properties of normal operators to a wider range of operators. One such notion is that of k-hyponormal operators. In this document, we focus our attention on the 2-hyponormality of weighted shift operators over a discrete Hilbert space. It will be shown that if a certain relation between the weights α = ( α0, α1,...) of a weighted shift Wα is satisfied, then the 2-hyponormality of Wα implies the hyponormality of Wαm for any m = 2, 3,....

Analysis Of The Cnn Algorithm In Target Recognition By Using The Mstar Database, Ligang Zou

Theses and Dissertations

With the rapid development of artificial intelligence technology and the emergence of a large number of innovative theories, the concept of deep learning is widely used in object detection, speech recognition, language translation and other fields. One of the important practices is target recognition in SAR images. Although it shows certain effectiveness in some researches, when using deep learning algorithm, there are still many problems that have not yet been solved. For example, people do not have a good understanding of how convolution works and the impact of convolution on the algorithm, although convolution works well in the CNN algorithm. …

Empirical Bayes Estimators And Borel-Tanner Distribution, Celestina Ruby Soltero

Theses and Dissertations

The motivation for this paper stems from the role Borel-Tanner (BT) distribution has as the distribution of the total outbreak number in epidemics modeled by branching processes. We briefly review Borel-Tanner distribution and its applications. In Chapter II we outline the Bayes decision problem, a construction for an Empirical Bayes (EB) estimator proposed by Liang [9] and discuss risk analysis. In Chapter III, the importance of randomization addressed and a classical construction of a monotonized EB estimator proposed by Houwalingen [14] is outlined. Lastly in Chapter IV we use R software to perform a Monte Carlo simulation and conduct a …

Extensions Of The Power Group Enumeration Theorem, Shawn Jeffrey Green

Theses and Dissertations

The goal of this paper is to develop extensions of Polya enumeration methods which count orbits of functions. De Bruijn, Harary, and Palmer all worked on this problem and created generalizations which involve permuting the codomain and domain of functions simultaneously. We cover their results and specifically extend them to the case where the group of permutations need not be a direct product of groups. In this situation, we develop a way of breaking the orbits into subclasses based on a characteristic of the functions involved. Additionally, we develop a formula for the number of orbits made up of bijective …

A Nonlinear Parallel Model For Reversible Polymer Solutions In Steady And Oscillating Shear Flow, Erik Tracey Palmer

Theses and Dissertations

A mathematical model for reversible polymers in steady and oscillating shear flows is presented. Using a mean-field approach, the behavior of the polymer network is characterized by a finitely extensible nonlinear elastic bead-spring model that stochastically transitions between dumbbell states to represent attachments, detachments and loops. An efficient parallel scheme for computation on GPUs utilizes populations of over a million dumbbells to characterize steady, large and small amplitude oscillatory shear (SAOS) flows in Brownian dynamics simulations. In steady-shear a novel attachment species transition function enables shear thickening and shear thinning by the adjustment of either attachment or detachment parameters. Three …

An Implementation Of The Kapustin-Li Formula, Jessica Otis

Theses and Dissertations

Let R be a regular local ring and take ! to be an isolated singularity on R. Taking Z/2-graded R-modules, X and Y , a matrix factorization of ω is a pair of morphisms (ϕ, ψ) such that ϕ⃝◦ψ = ω and ψ⃝◦ϕ = ω are satisfied in the diagram X ϕ → Y ψ→ X. We will discuss the category of matrix factorizations of ! in R and lead into the homotopy category of matrix factorizations as well as its historical development. Finally, we will conclude with the statement …

Sequential Survival Analysis With Deep Learning, Seth William Glazier

Theses and Dissertations

Survival Analysis is the collection of statistical techniques used to model the time of occurrence, i.e. survival time, of an event of interest such as death, marriage, the lifespan of a consumer product or the onset of a disease. Traditional survival analysis methods rely on assumptions that make it difficult, if not impossible to learn complex non-linear relationships between the covariates and survival time that is inherent in many real world applications. We first demonstrate that a recurrent neural network (RNN) is better suited to model problems with non-linear dependencies in synthetic time-dependent and non-time-dependent experiments.

A Few Problems On The Steiner Distance And Crossing Number Of Graphs, Josiah Reiswig

Theses and Dissertations

We provide a brief overview of the Steiner ratio problem in its original Euclidean context and briefly discuss the problem in other metric spaces. We then review literature in Steiner distance problems in general graphs as well as in trees.

Given a connected graph G we examine the relationship between the Steiner k-diameter, sdiam_k(G), and the Steiner k-radius, srad_k(G). In 1990, Henning, Oellermann and Swart [Ars Combinatoria 12 13-19, (1990)] showed that for any connected graph G, sdiam₃(G) ≤(8/5)srad₃(G) and …

Unconditionally Energy Stable Linear Schemes For A Two-Phase Diffuse Interface Model With Peng-Robinson Equation Of State, Chenfei Zhang

Theses and Dissertations

Many problems in the fields of science and engineering, particularly in materials science and fluid dynamic, involve flows with multiple phases and components. From mathematical modeling point of view, it is a challenge to perform numerical simulations of multiphase flows and study interfaces between phases, due to the topological changes, inherent nonlinearities and complexities of dealing with moving interfaces.

In this work, we investigate numerical solutions of a diffuse interface model with Peng-Robinson equation of state. Based on the invariant energy quadratization approach, we develop first and second order time stepping schemes to solve the liquid-gas diffuse interface problems for …

Finding Resolutions Of Mononomial Ideals, Hannah Melissa Kimbrell

Theses and Dissertations

In this paper we present two different combinatorial approaches to finding resolutions of polynomial ideals. Their goal is to get resolutions that are as small as possible while still preserving the structure of the zeroth syzygy module. Then we present the idea of a differential graded algebra and discuss when the minimal resolutions of a polynomial ideals admits such a structure.

Reduction Of The Kp Hierarchy, Adrian Eugenio Torres

Theses and Dissertations

This thesis will delve into the Kadomtsev-Petviashvili equation or KP equation and it's hierarchy. More specifically, the solition theory around it. To do so, we first explore the soliton theory for the Korteweg de-Vries equation or KdV equation by analysing it through the inverse scattering transform method and presenting it's soliton solutions. Second, we will introduce, Hirota's bilinear form, and understand its main idea. Third, introduce Sato Theory, and use it to formulate the KP hierarchy, via using pseudo-differential operators, presenting the lax operator, the dressing operator, Sato’s equation, and the zero curvature equation (Zakharov-Shabat Equation). Fourth, find the general …

Developing Understanding Of The Chain Rule, Implicit Differentiation, And Related Rates: Towards A Hypothetical Learning Trajectory Rooted In Nested Multivariation, Haley Paige Jeppson

Theses and Dissertations

There is an overemphasis on procedures and manipulation of symbols in calculus and not enough emphasis on conceptual understanding of the subject. Specifically, students struggle to understand and correctly apply concepts in calculus such as the chain rule, implicit differentiation, and related rates. Students can learn mathematics more deeply when they make connections between different mathematical ideas. I have hypothesized that students can make powerful connections between the chain rule, implicit differentiation, and related rates through the mathematical concept of nested multivariation. Based on this hypothesis, I created a hypothetical learning trajectory (HLT) rooted in nested multivariation for students to …

Hyperparameters For Dense Neural Networks, Christopher James Hettinger

Theses and Dissertations

Neural networks can perform an incredible array of complex tasks, but successfully training a network is difficult because it requires us to minimize a function about which we know very little. In practice, developing a good model requires both intuition and a lot of guess-and-check. In this dissertation, we study a type of fully-connected neural network that improves on standard rectifier networks while retaining their useful properties. We then examine this type of network and its loss function from a probabilistic perspective. This analysis leads to a new rule for parameter initialization and a new method for predicting effective learning …