Digital Commons Network^™

Problems In Chemical Graph Theory Related To The Merrifield-Simmons And Hosoya Topological Indices, William B. O'Reilly

College of Graduate Studies: Theses & Dissertations

In some sense, chemical graph theory applies graph theory to various physical sciences. This interdisciplinary field has significant applications to structure property relationships, as well as mathematical modeling. In particular, we focus on two important indices widely used in chemical graph theory, the Merrifield-Simmons index and Hosoya index. The Merrifield-Simmons index and the Hosoya index are two well-known topological indices used in mathematical chemistry for characterizing specific properties of chemical compounds. Substantial research has been done on the two indices in terms of enumerative problems and extremal questions. In this thesis, we survey known extremal results and consider the generalized …

A Corona Theorem For Multipliers On The Dirichlet Space, Alea Wittig

Electronic Theses & Dissertations (2024 - present)

An analogue to Wolff's ideal problem for the multiplier algebra of the Dirichlet space, the main theorem provides sufficient conditions to classify membership of an arbitrary function in an infinitely generated ideal of the multiplier algebra.

Snow Spheres And Ice Cubes: Solutions For Fermi Questions, February 2024, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Utility In Time Description In Priority Best-Worst Discrete Choice Models: An Empirical Evaluation Using Flynn's Data, Sasanka Adikari, Norou Diawara

Mathematics & Statistics Faculty Publications

Discrete choice models (DCMs) are applied in many fields and in the statistical modelling of consumer behavior. This paper focuses on a form of choice experiment, best-worst scaling in discrete choice experiments (DCEs), and the transition probability of a choice of a consumer over time. The analysis was conducted by using simulated data (choice pairs) based on data from Flynn's (2007) 'Quality of Life Experiment'. Most of the traditional approaches assume the choice alternatives are mutually exclusive over time, which is a questionable assumption. We introduced a new copula-based model (CO-CUB) for the transition probability, which can handle the dependent …

Infusing Machine Learning And Computational Linguistics Into Clinical Notes, Funke V. Alabi, Onyeka Omose, Omotomilola Jegede

Mathematics & Statistics Faculty Publications

Entering free-form text notes into Electronic Health Records (EHR) systems takes a lot of time from clinicians. A large portion of this paper work is viewed as a burden, which cuts into the amount of time doctors spend with patients and increases the risk of burnout. We will see how machine learning and computational linguistics can be infused in the processing of taking clinical notes. We are presenting a new language modeling task that predicts the content of notes conditioned on historical data from a patient's medical record, such as patient demographics, lab results, medications, and previous notes, with the …

Sparse Representer Theorems For Learning In Reproducing Kernel Banach Spaces, Rui Wang, Yuesheng Xu, Mingsong Yan

Mathematics & Statistics Faculty Publications

Sparsity of a learning solution is a desirable feature in machine learning. Certain reproducing kernel Banach spaces (RKBSs) are appropriate hypothesis spaces for sparse learning methods. The goal of this paper is to understand what kind of RKBSs can promote sparsity for learning solutions. We consider two typical learning models in an RKBS: the minimum norm interpolation (MNI) problem and the regularization problem. We first establish an explicit representer theorem for solutions of these problems, which represents the extreme points of the solution set by a linear combination of the extreme points of the subdifferential set, of the norm function, …

A Copula Discretization Of Time Series-Type Model For Examining Climate Data, Dimuthu Fernando, Olivia Atutey, Norou Diawara

Mathematics & Statistics Faculty Publications

The study presents a comparative analysis of climate data under two scenarios: a Gaussian copula marginal regression model for count time series data and a copula-based bivariate count time series model. These models, built after comprehensive simulations, offer adaptable autocorrelation structures considering the daily average temperature and humidity data observed at a regional airport in Mobile, AL.

Pigeon Strike!, John Adam

Mathematics & Statistics Faculty Publications

This article, titled "Pigeon strike!" from the Physics Teacher journal, discusses an incident where a bird, likely a pigeon, struck a window leaving an imprint. The author went outside to check but found no trace of the bird, hoping it had flown away after receiving medical care. The article also includes questions for readers to estimate the momentum, kinetic energy, force, pressure, and wingspan of the bird. The answers to these questions can be found online in the current issue of the journal. The article concludes with information about Fermi Questions and how to submit ideas.

The Weighty Words Of Lord Rayleigh: Solutions For Fermi Questions, January 2024, John Adam

Mathematics & Statistics Faculty Publications

No abstract provided.

Sky Smiley Face, John Adam

Mathematics & Statistics Faculty Publications

The article discusses the formation of a circumzenithal arc (CZA) in the sky, caused by specific light ray paths through hexagonal ice crystals in cirrus clouds. The CZA is visible when the Sun's altitude is about 32° or less. The text includes questions for readers to explore the physics behind the formation of CZAs and circumhorizontal arcs (CHAs). The author acknowledges photographers for providing images and invites readers to find answers online.

Earthquakes And Seismic Waves: Solutions For Fermi Questions, March 2024, John Adam

Mathematics & Statistics Faculty Publications

Solution to Question 1: To estimate h, we take the geometric mean of 1 km and 70 km, or ~8 km, and take ρ ≈ 2500 kg/m³.

Hence, δl ≈ 4 × 0.05 × (2.5 × 10³ kg/m³) × (10 m/s²) × (8 × 10³ m) × (2 × 10⁴ m) × 0.1 ÷ (3 × 10¹⁰ kg · m⁻¹ · s⁻²) ≈ 1.3 m. (Note that this is not meant to be an estimate of the width of the Icelandic rock fracture in the photograph, though it appears to be of the right order …

Earthquakes And Seismic Waves, John Adam

Mathematics & Statistics Faculty Publications

This article discusses earthquakes and seismic waves. It explains that earthquakes occur in the crust or upper mantle, with "shallow" earthquakes happening above a depth of 70 km in the crust. The article also introduces the slider-block model, which is a simple model used to understand earthquake behavior. It provides equations to calculate the displacement and maximum slip velocity of the fault based on various factors. The article then presents two questions related to earthquake calculations. Additionally, it mentions Fermi Questions, which are brief questions with estimation techniques.

Cats, Dogs, And Roll Waves: Solutions For Fermi Questions, April 2024, John Adam

Mathematics & Statistics Faculty Publications

On a hot afternoon in western North Carolina, the heavens opened and (to mix metaphors) it started raining “cats and dogs.” The rain was torrential, and soon a river of water was flowing downhill into the parking lot. In rivers and dam spillways, these are known as flood waves. The picture shows a particular type of flood waves known s roll waves—shock-like patterns separated by smooth profiles. In a steady-state situation, the drag forces on the water in the channel are balanced by the down-slope gravitational force. The dimensionless friction coefficient is C, the stream depth is h, v is …

Cats, Dogs, And Roll Waves, John Adam

Mathematics & Statistics Faculty Publications

This article discusses the phenomenon of roll waves, which are shock-like patterns separated by smooth profiles that occur in flood waves. The author mentions that in a steady-state situation, the drag forces on the water in the channel are balanced by the down-slope gravitational force. The article also includes questions for readers to estimate the speed and length of the roll waves, as well as determine the stability of the flow. The author provides additional information about the Chezy formula, an empirical equation used in fluid mechanics to estimate the mean flow velocity of open channel flow.

Recommendations To Internal Auditors Regarding The Auditing And Attestation Of Mathematical Programming Models, Jose Rincón, Greg Akai, Daryl Ono

Librarian Publications & Presentations

Mathematical programming planning models increase operational efficiency and minimize operating costs, but the underlying mathematics generally is complex. Combinatorial optimization is technically sophisticated which requires a strong quantitative background to successfully implement. Most internal auditors will not have the technical training to critically assess the underlying mathematics of mathematical programming planning models, but the internal auditor can still provide insight and attestation which can increase the efficiency of mathematical programming planning models.

Continuous-Variable Quantum Computation Of The O(3) Model In 1+1 Dimensions, Raghav G. Jha, Felix Ringer, George Siopsis, Shane Thompson

Physics Faculty Publications

We formulate the O(3) nonlinear sigma model in 1+1 dimensions as a limit of a three-component scalar field theory restricted to the unit sphere in the large squeezing limit. This allows us to describe the model in terms of the continuous-variable (CV) approach to quantum computing. We construct the ground state and excited states using the coupled-cluster Ansatz and find excellent agreement with the exact diagonalization results for a small number of lattice sites. We then present the simulation protocol for the time evolution of the model using CV gates and obtain numerical results using a photonic quantum simulator. We …

Integrable Symplectic Maps With A Polygon Tessellation, T. Zolkin, Y. Kharkov, S. Nagaitsev

Physics Faculty Publications

Identifying integrable dynamics remains a formidable challenge, and despite centuries of research, only a handful of examples are known to date. In this article, we explore a distinct form of area-preserving (symplectic) mappings derived from the stroboscopic Poincaré cross section of a kicked rotator—an oscillator subjected to an external force periodically switched on in short pulses. The significance of this class of problems extends to various applications in physics and mathematics, including particle accelerators, crystallography, and studies of chaos. Notably, Suris's theorem constrains the integrability within this category of mappings, outlining potential scenarios with analytic invariants of motion. In this …

Classification In Supervised Statistical Learning With The New Weighted Newton-Raphson Method, Toma Debnath

College of Graduate Studies: Theses & Dissertations

In this thesis, the Weighted Newton-Raphson Method (WNRM), an innovative optimization technique, is introduced in statistical supervised learning for categorization and applied to a diabetes predictive model, to find maximum likelihood estimates. The iterative optimization method solves nonlinear systems of equations with singular Jacobian matrices and is a modification of the ordinary Newton-Raphson algorithm. The quadratic convergence of the WNRM, and high efficiency for optimizing nonlinear likelihood functions, whenever singularity in the Jacobians occur allow for an easy inclusion to classical categorization and generalized linear models such as the Logistic Regression model in supervised learning. The WNRM is thoroughly investigated …

Enumeration Of Lattice Paths With Restrictions, Vince White

College of Graduate Studies: Theses & Dissertations

Lattice path enumeration, through the lens of Catalan numbers, plays a crucial role in combinatorics. This thesis delves into enumerations of some of the most common lattice paths – north-east paths, up-down paths, and Dyck paths – with restrictions applied. The first restriction is counting north-east lattice paths that only cross the diagonal line, y=x, once. The second form of lattice paths with restrictions is up-down paths that cross the x-axis exactly once and fall to a fixed depth of k. While working through this module, a novel proof for a known integer sequence was used, then applied to generate …

Counting Conjugates Of Colored Compositions, Jesus Omar Sistos Barron

Honors College Theses

The properties of n-color compositions have been studied parallel to those of regular compositions. The conjugate of a composition as defined by MacMahon, however, does not translate well to n-color compositions, and there is currently no established analogous concept. We propose a conjugation rule for cyclic n-color compositions. We also count the number of self-conjugates under these rules and establish a couple of connections between these and regular compositions.

Self-Exciting Point Processes In Real Estate, Ian Fraser

Theses and Dissertations (Comprehensive)

This thesis introduces a novel approach to analyzing residential property sales through the lens of stochastic processes by employing point processes. Herein, property sales are treated as point patterns, using self-exciting point process models and a variety of statistical tools to uncover underlying patterns in the data. Key findings include the identification and explanation of clustering in both space and time, and the efficacy of a temporal Hawkes process with a sinusoidal background in predicting home sale occurrences. The temporal analysis starts by employing the state of art techniques for time series data like regression, autoregressive, and autoregressive integrated moving …

Recoloring In Hereditary Graph Classes: Structure And Decomposition, Manoj Belavadi

Theses and Dissertations (Comprehensive)

In this thesis we study reconfiguration problems in graph theory. A reconfiguration problem is generally defined on the solution space of a problem for which a configuration can be defined as a feasible solution, for example, a coloring of a graph. In Chapters 1 through 4 we study the reconfiguration of vertex colorings. The reconfiguration graph of the k-colorings, denoted R_k(G), is the graph whose vertices are the k-colorings of G and two colorings are adjacent in R_k(G) if they differ on exactly one vertex. The basic question investigated here …

A Combinatorial Model For Affine Demazure Crystals Of Levels Zero And One, Samuel Spellman

Electronic Theses & Dissertations (2024 - present)

The symmetric and non-symmetric Macdonald polynomials are special families of orthogonal polynomials with parameters q and t. They are indexed by dominant, (resp. arbitrary) weights associated to a root system and generalize several well-known polynomials such as the Schur polynomials, Jack polynomials, Hall-Littlewood polynomials, etc. There are two well-known combinatorial models for computing these polynomials: a tableau model in type A, due to Haglund, Haiman and Loehr, and a type-independent model due to Ram and Yip, based on alcove walks.

Crystals bases are an important construction encoding information about Lie algebra representations. It turns out that there is an interesting …

Dynamics And Inverse Problems For Nonlinear Schrödinger Equations, Christopher Hogan

Doctoral Dissertations

"The cubic nonlinear Schrödinger equation (NLS) is a model of interest in the study of physical problems including nonlinear optics and Bose-Einstein condensates. Of particular interest is the study of cubic NLS with inhomogeneities such as localizations of the nonlinearity or terms introducing potential barriers. We first address some preliminaries and techniques useful in the study of the cubic NLS and its variations. We then consider the cubic NLS with a localized nonlinearity in dimensions d ≥ 2. We show that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that determines the X-ray transform of the …

The Deep Bsde Method, Daniel Kovach

Masters Theses

"The curse of dimensionality is the non-linear growth in computing time as the dimension of a problem increases. Using the Deep Backwards Stochastic Differential Equation (Deep BSDE) method developed in [HJE18], I approximate the solution at an initial time to a one-dimensional diffusion equation. Although we only approximate a one-dimensional equation, this method extends well to higher dimensions because it overcomes the curse of dimensionality by evaluating the given partial differential equation along "random characteristics''. In addition to the implementation, I also present most of the mathematical theory needed to understand this method"-- Abstract, p. iii

The Coulomb Gauge In Non-Associative Gauge Theory, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

The aim of this paper is to extend existence results for the Coulomb gauge from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogs of Lie groups. The main components of the theory include a finite-dimensional smooth loop L , its tangent algebra l , a finite-dimensional Lie group Ψ , that is the pseudoautomorphism group of L , a smooth manifold M with a principal Ψ -bundle P , and associated bundles Q and A with fibers L and l , respectively. A configuration in this theory is …

Symmetries And Integrable Systems, Sen-Yue Lou, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

Symmetry plays key roles in modern physics especially in the study of integrable systems because of the existence of infinitely many local and nonlocal generalized symmetries. In addition to the fundamental role to find exact group invariant solutions via Lie point symmetries, some important new developments on symmetries and conservation laws are reviewed. The recursion operator method is important to find infinitely many local and nonlocal symmetries of (1+1)-dimensional integrable systems. In this paper, it is pointed out that a recursion operator may be obtained from one key symmetry, say, a residual symmetry. For (2+1)-dimensional integrable systems, the master-symmetry approach …

Modeling The Effect Of Observational Social Learning On Parental Decision-Making For Childhood Vaccination And Diseases Spread Over Household Networks, Tamer Oraby, Andras Balogh

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we introduce a novel model for parental decision-making about vaccinations against a childhood disease that spreads through a contact network. This model considers a bilayer network comprising two overlapping networks, which are either Erdős–Rényi (random) networks or Barabási–Albert networks. The model also employs a Bayesian aggregation rule for observational social learning on a social network. This new model encompasses other decision models, such as voting and DeGroot models, as special cases. Using our model, we demonstrate how certain levels of social learning about vaccination preferences can converge opinions, influencing vaccine uptake and ultimately disease spread. In addition, …

Conditional Constrained And Unconstrained Quantization For Probability Distributions, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we present the idea of conditional quantization for a Borel probability measure P on a normed space Rk. We introduce the concept of conditional quantization in both constrained and unconstrained scenarios, along with defining the conditional quantization errors, dimensions, and coefficients in each case. We then calculate these values for specific probability distributions. Additionally, we demonstrate that for a Borel probability measure, the lower and upper quantization dimensions and coefficients do not depend on the conditional set of the conditional quantization in both constrained and unconstrained quantization.

Structure Of Fine Selmer Groups In Abelian P-Adic Lie Extensions, Debanjana Kundu, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio, Sujatha Ramdorai

School of Mathematical & Statistical Sciences Faculty Publications

This paper studies fine Selmer groups of elliptic curves in abelian p -adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Z p -extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.