Mathematics Commons^™

Data Assimilation For Conductance-Based Neuronal Models, Matthew Moye

Dissertations

This dissertation illustrates the use of data assimilation algorithms to estimate unobserved variables and unknown parameters of conductance-based neuronal models. Modern data assimilation (DA) techniques are widely used in climate science and weather prediction, but have only recently begun to be applied in neuroscience. The two main classes of DA techniques are sequential methods and variational methods. Throughout this work, twin experiments, where the data is synthetically generated from output of the model, are used to validate use of these techniques for conductance-based models observing only the voltage trace. In Chapter 1, these techniques are described in detail and the …

A Dynamic F5 Algorithm, Candice Mitchell

Dissertations

Gröbner bases are a “nice” representation for nonlinear systems of polynomials, where by “nice” we mean they have good computation properties. They have many useful applications, including decidability (whether the system has a solution or not), ideal membership (whether a given polynomial is in the system or not), and cryptography. Traditional Gröbner basis algorithms require as input an ideal and an admissible term ordering. They then determine a Gröbner basis with respect to the given ordering. Some term orderings lead to a smaller basis, but finding them traditionally requires testing many orderings and hoping for better results. A dynamic algorithm …

Extremal Problems On Induced Graph Colorings, James Hallas

Dissertations

Graph coloring is one of the most popular areas of graph theory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. As far back as 1880, in an attempt to solve the famous Four Color Problem, there have been numerous examples of certain types of graph colorings that have generated other graph colorings of interest. These types of colorings only gained momentum a century later, however, when in the 1980s, edge colorings were studied that led to vertex colorings of various types, led by the introduction …

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 …

Scalable Time-Stepping For Navier-Stokes Through High-Frequency Analysis Of Block Arnoldi Iteration, Brianna Bingham

Dissertations

Existing time-stepping methods for PDEs such as Navier-Stokes equations are not as efficient or scalable as they need to be for high-resolution simulation due to stiffness. The failure of existing time-stepping methods to adapt to changes in technology presents a dilemma that is becoming even more problematic over time. By rethinking approaches to time-stepping, dramatic gains in efficiency of simulation methods can be achieved. Krylov subspace spectral (KSS) methods have proven to be effective for solving time-dependent, variable-coefficient PDEs. The objective of this research is to continue the development of KSS methods to provide numerical solution methods that are far …

Variations In Ramsey Theory, Drake Olejniczak

Dissertations

The Ramsey number R(F,H) of two graphs F and H is the smallest positive integer n for which every red-blue coloring of the (edges of a) complete graph of order n results in a graph isomorphic to F all of whose edges are colored red (a red F) or a blue H. Beineke and Schwenk extended this concept to a bipartite version of Ramsey numbers, namely the bipartite Ramsey number BR(F,H) of two bipartite graphs F and H is the smallest positive integer rsuch that every red-blue coloring of the r-regular complete bipartite graph results in either …

Uniformly Connected Graphs, Nasreen Almohanna

Dissertations

Perhaps the most fundamental property that a graph can possess is that of being connected. Two vertices u and v of a graph G are connected if G contains a u-v path. The graph G itself is connected if every two vertices of G are connected. The well-studied concept of connectivity provides a measure on how strongly connected a graph may be. There are many other degrees of connectedness for a graph. A Hamiltonian path in a graph G is a path containing every vertex of G. Among the best-known classes of highly connected graph are the Hamiltonian-connected graphs, …

The Role Of Sampling Variability In Developing K-8 Preservice Teachers’ Informal Inferential Reasoning, Omar Abu-Ghalyoun

Dissertations

Recent influential policy reports, such as the Common Core State Standards (CCSS-M, 2010) and Guidelines for Assessment and Instruction in Statistics Education Report, (GAISE, 2007), have called for dramatic changes in the statistics content included in the K-8 curriculum. In particular, students in these grades are now expected to develop Informal Inferential Reasoning (IIR) as a way of preparing them for formal concepts of inferential statistics such as confidence intervals and testing hypotheses. Ben-Zvi, Gil, & Apel, (2007) describe IIR as the cognitive activities involved in informally making statistical inferences. Over this path from informal to formal inference, many important …

Domain Decomposition Methods For The Solution Of Multiple Scattering Problems, Michael Pedneault

Dissertations

This presents a Schur complement Domain Decomposition (DD) algorithm for the solution of frequency domain multiple scattering problems. Just as in the classical DD methods,(1) the ensemble of scatterers is enclosed in a domain bounded by an artificial boundary, (2) this domain is subdivided into a collection of nonoverlapping subdomains so that the boundaries of the subdomains do not intersect any of the scatterers, and (3) the solutions of the subproblems are connected via Robin boundary conditions matching on the common interfaces between subdomains. Subdomain Robin-to-Robin maps are used to recast the DD problem as a sparse linear system whose …

Fwer Controlling Procedures In Simultaneous And Selective Inference, Li Yu

Dissertations

With increasing complexity of research objectives in clinical trials, a variety of relatively complex and less intuitive multiple testing procedures (MTPs) have been developed and applied in clinical data analysis. In order to make testing strategies more explicit and intuitive to communicate with non-statisticians, several flexible and powerful graphical approaches have recently been introduced in the literature for developing and visualizing newer MTPs. Nevertheless, some theoretical as well as methodological issues still remain to be fully addressed. This dissertation addresses several important issues arising in graphical approaches and related selective inference problems. It consists of three parts.

In the first …

Predicted Deepwater Bathymetry From Satellite Altimetry: Non-Fourier Transform Alternatives, Maxsimo Salazar

Dissertations

Robert Parker (1972) demonstrated the effectiveness of Fourier Transforms (FT) to compute gravitational potential anomalies caused by uneven, non-uniform layers of material. This important calculation relates the gravitational potential anomaly to sea-floor topography. As outlined by Sandwell and Smith (1997), a six-step procedure, utilizing the FT, then demonstrated how satellite altimetry measurements of marine geoid height are inverted into seafloor topography. However, FTs are not local in space and produce Gibb’s phenomenon around discontinuities. Seafloor features exhibit spatial locality and features such as seamounts and ridges often have sharp inclines. Initial tests compared the windowed-FT to wavelets in reconstruction of …

Generalized Line Graphs, Mohra Abdullah Z. Alqahtani

Dissertations

With every nonempty graph, there are associated many graphs. One of the best known and most studied of these is the line graph L (G) of a graph G, whose vertices are the edges of G and where two vertices of L (G) are adjacent if the corresponding edges of G are adjacent. This concept was implicitly introduced by Whitney in 1932. Over the years, characterizations of graphs that are line graphs have been given, as well as graphs whose line graphs have some specified property. For example, Beineke characterized graphs that are line graphs by forbidding certain graphs …

Probabilistic And Extremal Problems In Combinatorics, Sean English

Dissertations

Graph theory as a mathematical branch has been studied rigorously for almost three centuries. In the past century, many new branches of graph theory have been proposed. One important branch of graph theory involves the study of extremal graph theory. In 1941, Turán studied one of the first extremal problems, namely trying to maximize the number of edges over all graphs which avoid having certain structures. Since then, a large body of work has been created in the study of similar problems. In this dissertation, a few different extremal problems are studied, but for hypergraphs rather than graphs. In particular, …

Graceful Colorings And Connection In Graphs, Alexis D. Byers

Dissertations

For a graph G of size m, a graceful labeling of G is an injective function f : V (G) → {0, 1, . . . , m} that gives rise to a bijective function f ¹ : E(G) → {1, 2, . . . , m} defined by f ¹(uv) = |f (u) − f (v)|. A graph is graceful if it has a graceful labeling. Over the years, a number of variations of graceful …

The Bellringer Sequence: Investigating What And How Preservice Mathematics Teachers Learn Through Pedagogies Of Enactment, Mary A. Ochieng

Dissertations

This study examines preservice teacher learning through pedagogies of enactment—approaches to teacher education that allow preservice teachers to learn by doing what teachers do. Preservice teacher (PST) learning is examined through the implementation of the Bellringer Sequence (BRS), a pedagogy of enactment conceptualized in the study. The BRS is centered around bellringers—brief mathematical tasks implemented as students arrive for class. The BRS is a sequence of four activities centered on a bellringer: preparation (for teaching a bellringer) implementation (of the bellringer with peers), debriefing (discussing the implementation as colleagues), and written reflection (about the effectiveness of the bellringer).

Practice-based approaches …

Induced Graph Colorings, Ian Hart

Dissertations

An edge coloring of a nonempty graph G is an assignment of colors to the edges of G. In an unrestricted edge coloring, adjacent edges of G may be colored the same. If every two adjacent edges of G are colored differently, then this edge coloring is proper and the minimum number of colors in a proper edge coloring of G is the chromatic index χ^/(G) of G. A proper vertex coloring of a nontrivial graph G is an assignment of colors to the vertices of G such that every two adjacent vertices of …

Survival Analysis Using Archimedean Copulas, Xieyang Jia

Dissertations

This dissertation has three independent parts. The first part studies a variation of the competing risks problem, known as the semi-competing risks problem, in which a terminal event censors a non-terminal event, but not vice versa, in the presence of a censoring event which is independent of these two events. The joint distribution of the two dependent events is formulated under Archimedean copula. An estimator for the association parameter of the copula is proposed, which is shown to be consistent. Simulation shows that the method works well with most common Archimedean copula models.

The second part studies the properties of …

Numerical Simulations Of Thin Viscoelastic Films, Valeria Barra

Dissertations

This dissertation is developed in the field of Computational Fluid Dynamics (CFD) and it focuses on numerical simulations of the dynamics of thin viscoelastic films in different settings. The first part of this dissertation presents a novel computational investigation of thin viscoelastic films and drops, that are subject to the van der Waals interaction force, in two spatial dimensions. The liquid films are deposited on a flat solid substrate, that can have a zero or nonzero inclination with respect to the base. The equation that governs the interfacial dynamics of the thin films and drops is obtained within the long-wave …

Instabilities In Nematic Liquid Crystal Films And Droplets, Michael-Angelo Y.-H. Lam

Dissertations

The dynamics of thin films of nematic liquid crystal (NLC) are studied. Nematic liquid crystals are a type of non-Newtonian fluid with anisotropic viscous effects (due to the shape of the molecules) and elasticity effects (due to interacting electrical dipole moments). Exploiting the small aspect ratio in the geometry of interest, a fourth-order non-linear partial differential equation is used to model the free surface of the thin films. Particular attention is paid to the interplay between the bulk elasticity and the preferred orientation (boundary condition) of NLC molecules at the two interfaces: the substrate and the free surface. This work …

Rapid Generation Of Jacobi Matrices For Measures Modified By Rational Factors, Amber Sumner

Dissertations

Orthogonal polynomials are important throughout the fields of numerical analysis and numerical linear algebra. The Jacobi matrix J for a family of n orthogonal polynomials is an n x n tridiagonal symmetric matrix constructed from the recursion coefficients for the three-term recurrence satisfied by the family. Every family of polynomials orthogonal with respect to a measure on a real interval [a,b] satisfies such a recurrence. Given a measure that is modified by multiplying by a rational weight function r(t), an important problem is to compute the modified Jacobi matrix Jmod corresponding to the new measure from knowledge of J. There …

Edge Induced Weightings Of Uniform Hypergraphs And Related Problems, Laars C. Helenius

Dissertations

The starting point of the research is the so called 1-2-3 Conjecture formulated in 2004 by Karoński, Luczak, and Thomason. Roughly speaking it says that the edges of any graph can be weighted from {1, 2, 3} so that the induced vertex coloring (as the sum of weights adjacent to a given vertex) is proper. The conjecture has attracted a lot of interest from researchers over the last decade but is still unanswered. More recently, the conjecture has been studied for hypergraphs.

The main result of this dissertation shows in particular that an analogous conjecture holds for almost all uniform …

Numerical Methods And Simulation For Time Dependent Electrokinetic Flow, Rui Cao

Dissertations

Electrokinetic flow typically occurs when an electric field is applied to a fluid electrolyte in the presence of a boundary or interface. The electric field induces a force that causes oppositely charged ions to migrate in opposite directions while simultaneously diffusing due to Brownian motion. In an unbounded medium this process, which is called electrodiffusion, does not separate bulk electric charge into separate regions and causes no bulk motion of the host fluid solvent. However, in the presence of a boundary or interface that either impedes or is impervious to the movement of ions in the normal direction, the normal …

Mathematical Models For Polymer-Nematic Interactions, Ensela Mema

Dissertations

This dissertation considers a mathematical model that consists of a nematic liquid crystal layer sandwiched between two parallel bounding plates, across which an external field may be applied. Particular attention is paid to the effect of an applied field on the layer as well as the interaction between the liquid crystal molecules and the molecules of the substrate. The system studied may be considered as a simple model of a Liquid Crystal Display (LCD) device, and the results obtained are discussed and interpreted within this context.

The first part of this dissertation considers a study that investigates how the number …

Topics On Multiple Hypotheses Testing And Generalized Linear Model, Yalin Zhu

Dissertations

In applications such as studying drug adverse events (AE) in clinical trials and identifying differentially expressed genes in microarray experiments, the data of the experiments usually consists of frequency counts. In the analysis of such data, researchers often face multiple hypotheses testing based on discrete test statistics. Incorporating this discrete property of the data, several stepwise procedures, which allow to use the CDF of p-values to determine the testing threshold, are proposed for controlling familiwise error rate (FWER). It is shown that the proposed procedures strongly control the FWER and are more powerful than the existing ones for discrete data. …

Direct Computations Of Marangoni Driven Flows Using A Volume Of Fluid Method, Ivana Seric

Dissertations

The volume of fluid (VoF) interface tracking methods have been used for simulating a wide range of interfacial flows. An improved accuracy of the surface tension force computation has enabled the VoF method to become widely used for simulating flows driven by the surface tension force. A general methodology for the inclusion of variable surface tension coefficient into a VoF based Navier-Stokes solver is developed. This new numerical model provides a robust and accurate method for computing the surface gradients directly by finding the tangent directions on the interface using height functions. The implementation applies to both temperature and concentration …

Solution Of Pdes For First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods, Somayyeh Sheikholeslami

Dissertations

We solve the first order reaction-diffusion equations which describe binding-diffusion kinetics using a photobleaching scanning profile of a confocal laser scanning microscope approximated by a Gaussian laser profile. We show how to solve these equations with prebleach steady-state initial conditions using a time-domain method known as a Krylov Subspace Spectral (KSS) method. KSS methods are explicit methods for solving time- dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of their stability and their superior scalability. These advantages are obtained by applying Gaussian quadrature rules in the spectral domain developed by Golub and Meurant. …

Efficient Coarse-Grained Brownian Dynamics Simulations For Dna And Lipid Bilayer Membrane With Hydrodynamic Interactions, Szu-Pei Fu

Dissertations

The coarse-grained molecular dynamics (CGMD) or Brownian dynamics (BD) simulation is a particle-based approach that has been applied to a wide range of biological problems that involve interactions with surrounding fluid molecules or the so-called hydrodynamic interactions (HIs). From simple biological systems such as a single DNA macromolecule to large and complicated systems, for instances, vesicles and red blood cells (RBCs), the numerical results have shown outstanding agreements with experiments and continuum modeling by adopting Stokesian dynamics and explicit solvent model. Finally, when combined with fast algorithms such as the fast multipole method (FMM) which has nearly optimal complexity in …

Boundary Integral Equation Based Numerical Solutions Of Helmholtz Transmission Problems For Composite Scatters, Haiyang Qi

Dissertations

In this dissertation, an in-depth comparison between boundary integral equation solvers and Domain Decomposition Methods (DDM) for frequency domain Helmholtz transmission problems in composite two-dimensional media is presented. Composite media are characterized by piece-wise constant material properties (i.e., index of refraction) and thus, they exhibit interfaces of material discontinuity and multiple junctions. Whenever possible to use, boundary integral methods for solution of Helmholtz boundary value problems are computationally advantageous. Indeed, in addition to the dimensional reduction and straightforward enforcement of the radiation conditions that these methods enjoy, they do not suffer from the pollution effect present in volumetric discretization. The …

Highly Hamiltonian Graphs And Digraphs, Zhenming Bi

Dissertations

A cycle that contains every vertex of a graph or digraph is a Hamiltonian cycle. A graph or digraph containing such a cycle is itself called Hamiltonian. This concept is named for the famous Irish physicist and mathematician Sir William Rowan Hamilton. These graphs and digraphs have been the subject of study for over six decades. In this dissertation, we study graphs and digraphs with even stronger Hamiltonian properties, namely highly Hamiltonian graphs and digraphs.