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Convex Relaxations Of A Continuum Aggregation Model, And Their Efficient Numerical Solution, Mahdi Bandegi

Dissertations

In this dissertation, the global minimization of a large deviations rate function (the Helmholtz free energy functional) for the Boltzmann distribution is discussed. The Helmholtz functional arises in large systems of interacting particles — which are widely used as models in computational chemistry and molecular dynamics. Global minimizers of the rate function (Helmholtz functional) characterize the asymptotics of the partition function and thereby determine many important physical properties such as self-assembly, or phase transitions. Finding and verifying local minima to the Helmholtz free energy functional is relatively straightforward. However, finding and verifying global minima is much more difficult since the …

Dimension Reduction Techniques For High Dimensional And Ultra-High Dimensional Data, Subha Datta

Dissertations

This dissertation introduces two statistical techniques to tackle high-dimensional data, which is very commonplace nowadays. It consists of two topics which are inter-related by a common link, dimension reduction.

The first topic is a recently introduced classification technique, the weighted principal support vector machine (WPSVM), which is incorporated into a spatial point process framework. The WPSVM possesses an additional parameter, a weight parameter, besides the regularization parameter. Most statistical techniques, including WPSVM, have an inherent assumption of independence, which means the data points are not connected with each other in any manner. But spatial data violates this assumption. Correlation between …

Model Selection And Experimental Design Of Biological Networks With Algebraic Geometry, Anyu Zhang

Mathematics Theses and Dissertations

Model selection based on experimental data is an essential challenge in biological data science. In decades, the volume of biological data from varied sources, including laboratory experiments, field observations, and patient health records has seen an unprecedented increase. Mainly when collecting data is expensive or time-consuming, as it is often in the case with clinical trials and biomolecular experiments, the problem of selecting information-rich data becomes crucial for creating relevant models.

Motivated by certain geometric relationships between data, we partitioned input data sets, especially data sets that correspond to a unique basis, into equivalence classes with the same basis to …

A Qualitative Representation Of Spatial Scenes In R2 With Regions And Lines, Joshua Lewis

Electronic Theses and Dissertations

Regions and lines are common geographic abstractions for geographic objects. Collections of regions, lines, and other representations of spatial objects form a spatial scene, along with their relations. For instance, the states of Maine and New Hampshire can be represented by a pair of regions and related based on their topological properties. These two states are adjacent (i.e., they meet along their shared boundary), whereas Maine and Florida are not adjacent (i.e., they are disjoint).

A detailed model for qualitatively describing spatial scenes should capture the essential properties of a configuration such that a description of the represented objects …

Theory Of Lexicographic Differentiation In The Banach Space Setting, Jaeho Choi

Electronic Theses and Dissertations

Derivative information is useful for many problems found in science and engineering that require equation solving or optimization. Driven by its utility and mathematical curiosity, researchers over the years have developed a variety of generalized derivatives. In this thesis, we will first take a look at Clarke’s generalized derivative for locally Lipschitz continuous functions between Euclidean spaces, which roughly is the smallest convex set containing all nearby derivatives of a domain point of interest. Clarke’s generalized derivative in this setting possesses a strong theoretical and numerical toolkit, which is analogous to that of the classical derivative. It includes nonsmooth versions …

Enriched Derivators, James Richardson

Electronic Thesis and Dissertation Repository

In homotopical algebra, the theory of derivators provides a convenient abstract setting for computing with homotopy limits and colimits. In enriched homotopy theory, the analogues of homotopy (co)limits are weighted homotopy (co)limits. In this thesis, we develop a theory of derivators and, more generally, prederivators enriched over a monoidal derivator E. In parallel to the unenriched case, these E-prederivators provide a framework for studying the constructions of enriched homotopy theory, in particular weighted homotopy (co)limits.

As a precursor to E-(pre)derivators, we study E-categories, which are categories enriched over a bicategory Prof(E) associated to E. We prove a number of fundamental …

Characterizing Compact Game Trees, Andrew Dubose

UNLV Theses, Dissertations, Professional Papers, and Capstones

It is well-known that the body of a game tree of height less than or equal to ω is compact

if and only if the tree is finitely branching. In this thesis, we develop necessary and sufficient

conditions for the body of any game tree to be compact.

Arbitrary High Order Finite Difference Methods With Applications To Wave Propagation Modeled By Maxwell's Equations, Min Chen

UNLV Theses, Dissertations, Professional Papers, and Capstones

This dissertation investigates two different mathematical models based on the time-domain Maxwell's equations: the Drude model for metamaterials and an equivalent Berenger's perfectly matched layer (PML) model. We develop both an explicit high order finite difference scheme and a compact implicit scheme to solve both models. We develop a systematic technique to prove stability and error estimate for both schemes. Extensive numerical results supporting our analysis are presented. To our best knowledge, our convergence theory and stability results are novel and provide the first error estimate for the high-order finite difference methods for Maxwell's equations.

A Pedagogic Analysis Of Linear Algebra Courses, Andrew Taylor

Mathematics & Statistics ETDs

This project is concerned with investigating the question, "Do our applied linear algebra courses (at the University of New Mexico) adequately prepare STEM students for future work in their respective fields?" In order to explore this, surveys were issued to three groups (sections) of students (among two different instructors) at the conclusion of their applied linear algebra course, as well as STEM professors/instructors from a variety of STEM fields. Students were surveyed regarding their perceived mastery of given topics/ideas from the course and professors/instructors were surveyed about the level of mastery they felt was necessary (referred to as ``desired mastery") …

Albert Forms, Quaternions, Schubert Varieties & Embeddability, Jasmin Omanovic

Electronic Thesis and Dissertation Repository

The origin of embedding problems can be understood as an effort to find some minimal datum which describes certain algebraic or geometric objects. In the algebraic theory of quadratic forms, Pfister forms are studied for a litany of powerful properties and representations which make them particularly interesting to study in terms of embeddability. A generalization of these properties is captured by the study of central simple algebras carrying involutions, where we may characterize the involution by the existence of particular elements in the algebra. Extending this idea even further, embeddings are just flags in the Grassmannian, meaning that their study …

The Negotiator's Role In A Buyer-Seller Game, Joseph Gaudy

Graduate Theses and Capstone Projects (excluding DNP)

In game theory, buyer-seller games rarely utilize a negotiating third party. Any negotiations are typically conducted by the buyer and seller. This study, motivated by the real estate market, uses sequentially and simultaneously played game models to explore the influence a self-interested, negotiating, third party has on player payoffs. For the sequential model, a game tree is utilized to demonstrate player actions, preferences, and outcomes. The weak sequential equilibrium is calculated using Gambit[1] and shows optimality in player payoffs to exist when the seller’s and realtor’s strategies align according to the current market. For the simultaneous model, expected payoff functions …

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

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 …

Three-Dimensional Analytical Model Of Tidal Flow In The Damariscotta River Estuary, Me, Stephanie L. Ayres

Electronic Theses and Dissertations

Estuaries are coastal bodies of water subjected to strong tidal influence and characterized by their morphology, tidal dynamics, topography, and stratification. Tidal flow is critically important to the water circulation, nutrient influx, and sediment transport in or out of an estuary. However, tidal asymmetry enhanced by estuary shape and nonlinear processes can lead to complications in estuarine flow. Analytical models are used to systematically study tidal flow within an estuary. Previous studies have derived analytical models of varying complexity and applied them to investigate tidal and residual flow. This thesis derives a three-dimensional analytical model with a perturbation expansion of …

Groups Satisfying The Converse To Lagrange's Theorem, Jonah N. Henry

MSU Graduate Theses

Lagrange’s theorem, which is taught early on in group theory courses, states that the order of a subgroup must divide the order of the group which contains it. In this thesis, we consider the converse to this statement. A group satisfying the converse to Lagrange’s theorem is called a CLT group. We begin with results that help show that a group is CLT, and explore basic CLT groups with examples. We then give the conditions to guarantee either CLT is satisfied or a non-CLT group exists for more advanced cases. Additionally, we show that CLT groups are properly contained between …

United States Suicide Analysis: 1999-2016, Malynn Clark

Honors Theses

The purpose of this thesis is to create information visualizations surrounding suicide trends from 1999-2016 in the United States. The original data was obtained from the Centers for Disease Control and Prevention’s Compressed Mortality Database. This database permits users to download several fields of information regarding deaths for the years given. Using this information, many graphs below show trends and patterns for suicide. One notable trend includes the higher proportion of male to female suicides for all categories explored including: age group, race, and metro/nonmetro status. The goal is to bring awareness and understanding surrounding the suicide epidemic in the …

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].

Implementation And Effects Of University College Algebra Growth Mindset Structured Assessments In Large Lectures, Hannah Mae Lewis

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Recent scientific evidence shows the incredible potential of the brain to grow and change. Students with a growth mindset view errors and obstacles as opportunities for growth. These students welcome challenges and the opportunity to learn from their mistakes. Although some university instructors are incorporating growth mindset into their lectures, attitudes, and exams in small classes, the traditional exam method used in large lecture undergraduate mathematics classrooms follows a fixed mindset model. The growth mindset structured assessments developed for this study incorporate a testing center portion (matching, short answer, fill in the blank and free response) with structured rework opportunities, …

Supporting The Algebra I Curriculum With An Introduction To Computational Thinking Course, Michelle M. Laskowski

LSU Master's Theses

The Louisiana Workforce Commission predicts a 33.6% increase in computer science and mathematical occupations by 2022 and the Bureau of Labor Statistics foresees a 16% increase in computer scientists from 2018-2028. Despite these opportunities for job and financial security, the number of Louisiana students enrolled in a nationally accredited computing course is less than 1%, compared to national leaders California and Texas which have 3% and 3.8% of students respectively. Furthermore, the international assessments of mathematical literacy, PISA and TIMMS, both report American students continue to fall further behind their international peers in mathematics achievement.

This thesis rejects these statistics …

Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen

Doctoral Dissertations

Let E₁ x E₂ over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E₁ x E₂, denoted as a_p(E₁) + a_p(E₂), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical …

Optimal Relaxation Weights For Multigrid Reduction In Time (Mgrit), Masumi Sugiyama

Mathematics & Statistics ETDs

Based on current trends in computer architectures, faster compute speeds must come from increased parallelism rather than increased clock speeds, which are stagnate. This situation has created the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. In this work, we consider the scheme known as multigrid-reduction-in-time (MGRIT), but note that there exist other parallel-in-time methods such as parareal and the parallel full approximation scheme in space and time (PFASST). MGRIT is a full multi-level method applied to the time dimension and …

Alpha Capture Reaction Rates For Nucleosynthesis Within An Ab Initio Framework, Alison Constance Dreyfuss

LSU Doctoral Dissertations

Clustering in nuclear systems has broad impacts on all phases of stellar burning, and plays a significant role in our understanding of nucleosynthesis, or how and where nuclei are produced in the universe. The role of alpha particles in particular is extremely important for nuclear astrophysics: ⁴He was one of the earliest elements produced in the Big Bang, it is one of the most abundant elements in the universe, and helium burning -- in particular, the triple-alpha process -- is one of the most important ``engines'' in stars. To better understand nucleosynthesis and stellar burning, then, it is important …

An Introduction To Shape Dynamics, Patrick R. Kerrigan

Physics

Shape Dynamics (SD) is a new fundamental framework of physics which seeks to remove any non-relational notions from its methodology. importantly it does away with a background space-time and replaces it with a conceptual framework meant to reflect direct observables and recognize how measurements are taken. It is a theory of pure relationalism, and is based on different first principles then General Relativity (GR). This paper investigates how SD assertions affect dynamics of the three body problem, then outlines the shape reduction framework in a general setting.

Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, Lian Duan

Doctoral Dissertations

In this thesis, we prove that, a selfdual 3-dimensional Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to 3-dimensional Galois representations with ground field not equal to Q. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

Doctoral Dissertations

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and n equals 3. Our first main …

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 …

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 …

Tick Control Methods For Amblyomma Americanum In Virginia: Applications And Modeling, Alexis Lynn White

Biological Sciences Theses & Dissertations

Tick-borne diseases continue to increase in the United States, and yet no comprehensive method of tick control currently exists. The lone star tick, Amblyomma americanum, is an aggressive human-biting tick and vector of several pathogens which effect both humans and other animals. Standard control methods do not work as well for A. americanum as they do for the more commonly studied blacklegged tick, Ixodes scapularis. TickBot, a tick-killing robot, is a potential method to control A. americanum that lures ticks to its path with carbon dioxide and the ticks die from contact with a permethrin-treated cloth that is …

Dynamics Of The Family Lambda Tan Z^2, Santanu Nandi

Dissertations, Theses, and Capstone Projects

We prove some topological properties of the dynamical plane ($z$-plane) and a combinatorial structure of the parameter plane of a holomorphic family of meromorphic maps $\lambda \tan z^2$. In the dynamical plane, we prove that there is no Herman ring and the Julia set is a Cantor set for the map when the parameter is in the central capture component. Julia set is connected for the maps when the parameters are in other hyperbolic components. In the parameter plane, I prove that the capture components are simply connected and there are always four hyperbolic shell components attached to a virtual …

Hermitian Maass Lift For General Level, An Hoa Vu

Dissertations, Theses, and Capstone Projects

For an imaginary quadratic field $K$ of discriminant $-D$, let $\chi = \chi_K$ be the associated quadratic character. We will show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $\chi$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space …