Applied Mathematics Commons^™

Vertex-Magic Graphs, Karissa Massud

Honors Program Theses and Projects

In this paper, we will study magic labelings. Magic labelings were first introduced by Sedláček in 1963 [3]. At this time, the labels on the graph were only assigned to the edges. In 1970, Kotzig and Rosa defined what are now known as edge-magic total labelings, where both the vertices and the edges of the graph are labeled. Following this in 1999, MacDougall, Miller, Slamin, and Wallis introduced the idea of vertex-magic total labelings. There are many different types of magic labelings. In this paper will focus on vertex-magictotal labelings.

The Role Of Surprise In Guessing Games, Justin Carpender

Honors Program Theses and Projects

In this thesis we will study the connection between game structure, surprise, and guessing strategies for these first two versions of a word guessing game. Our analysis will have three levels: one, a basic understanding of language and letter probabilities and the creation of programs that seek to use the structure of a game to efficiently guess words; two, an introduction of mathematical background and Information Theory; three, an analysis of the games and their corresponding guesses via a creative use of the key ideas of Information Theory, particularly, the concepts of surprise and entropy.

Vertex-Magic Total Labeling On G-Sun Graphs, Melissa Mejia

Honors Program Theses and Projects

Graph labeling is an immense area of research in mathematics, specifically graph theory. There are many types of graph labelings such as harmonious, magic, and lucky labelings. This paper will focus on magic labelings. Graph theorists are particularly interested in magic labelings because of a simple problem regarding tree graphs introduced in the 1990’s. The problem is still unsolved after almost thirty years. Researchers have studied magic labelings on other graphs in addition to tree graphs. In this paper we will consider vertex-magic labelings on G-sun graphs. We will give vertex-magic total labelings for ladder sun graphs and complete bipartite …

Spectral Analysis Of Multiscale Cultural Traits On Twitter, Chandler Squires, Nikhil Kunapuli, Yaneer Bar-Yam, Alfredo Morales

Northeast Journal of Complex Systems (NEJCS)

Understanding and mapping the emergence and boundaries of cultural areas is a challenge for social sciences. In this paper, we present a method for analyzing the cultural composition of regions via Twitter hashtags. Cultures can be described as distinct combination of traits which we capture via principal component analysis (PCA). We investigate the top 8 PCA components of an area including France, Spain, and Portugal, in terms of the geographic distribution of their hashtag composition. We also discuss relationships between components and the insights those relationships can provide into the structure of a cultural space. Finally, we compare the spatial …

Mathematical Models Yield Insights Into Cnns: Applications In Natural Image Restoration And Population Genetics, Ryan Cecil

Electronic Theses and Dissertations

Due to a rise in computational power, machine learning (ML) methods have become the state-of-the-art in a variety of fields. Known to be black-box approaches, however, these methods are oftentimes not well understood. In this work, we utilize our understanding of model-based approaches to derive insights into Convolutional Neural Networks (CNNs). In the field of Natural Image Restoration, we focus on the image denoising problem. Recent work have demonstrated the potential of mathematically motivated CNN architectures that learn both `geometric' and nonlinear higher order features and corresponding regularizers. We extend this work by showing that not only can geometric features …

Lorentzian Polynomials, Higher Hessians, And The Hodge-Riemann Property For Codimension Two Graded Artinian Gorenstein Algebras, Pedro Macias-Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

We study the Hodge-Riemann property (HRP) for graded Artinian Gorenstein (AG) algebras. We classify AG algebras in codimension two that have HRP in terms of higher Hessian matrices and positivity of Schur functions associated to certain rectangular partitions.

In this paper we introduce the Hodge Riemann property (HRP) on an arbitrary graded oriented Artinian Gorenstein (AG) algebra defined over R, and we give a criterion on the higher Hessian matrix of its Macaulay dual generator (Theorem 3.1). AG algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of complex manifolds, and the HRP is analogous to …

Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang

Department of Mathematics: Faculty Publications

We study sequences of Betti numbers (β^R_i (M)) of finite modules M over a complete intersection local ring, R. It is known that for every M the subsequence with even, respectively, odd indices i is eventually given by some polynomial in i. We prove that these polynomials agree for all R-modules if the ideal I^☐ generated by the quadratic relations of the associated graded ring of R satisfies height I^☐ ≥ codim R − 1, and that the converse holds when R is homogeneous and when codim R ≤ 4. Avramov, …

Low-Gain Integral Control For A Class Of Discrete-Time Lur’E Systems With Applications To Sampled-Data Control, Chris Guiver, Richard Rebarber, Stuart Townley

Department of Mathematics: Faculty Publications

We study low-gain (P)roportional (I)ntegral control of multivariate discrete-time, forced Lur’e systems to solve the output-tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low-gain PI controller to achieve exponential disturbance-to-state and disturbance-to-tracking-error stability in closed-loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input-to-state stability. The discrete-time theory facilitates a similar result for a continuous-time forced Lur’e system in feedback with sampled-data low-gain integral control. The theory …

Debiasing Cyber Incidents – Correcting For Reporting Delays And Under-Reporting, Seema Sangari

Doctor of Data Science and Analytics Dissertations

This research addresses two key problems in the cyber insurance industry – reporting delays and under-reporting of cyber incidents. Both problems are important to understand the true picture of cyber incident rates. While reporting delays addresses the problem of delays in reporting due to delays in timely detection, under-reporting addresses the problem of cyber incidents frequently under-reported due to brand damage, reputation risk and eventual financial impacts.

The problem of reporting delays in cyber incidents is resolved by generating the distribution of reporting delays and fitting modeled parametric distributions on the given domain. The reporting delay distribution was found to …

Robust Uncertainty Quantification With Analysis Of Error In Standard And Non-Standard Quantities Of Interest, Zachary Stevens

Mathematics & Statistics ETDs

This thesis derives two Uncertainty Quantification (UQ) methods for differential equations that depend on random parameters: (\textbf{i}) error bounds for a computed cumulative distribution function (\textbf{ii}) a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria. Both UQ approaches utilize adjoint-based \textit{a posteriori} error analysis in order to accurately estimate the error in samples of numerically approximated quantities of interest. The adaptive MLMC algorithm developed in this thesis relies on the adjoint-based error analysis to adaptively create meshes and accurately monitor a stopping criteria. This is in contrast to classical MLMC algorithms which employ either a …

Machine Learning Model Comparison And Arma Simulation Of Exhaled Breath Signals Classifying Covid-19 Patients, Aaron Christopher Segura

Mathematics & Statistics ETDs

This study compared the performance of machine learning models in classifying COVID-19 patients using exhaled breath signals and simulated datasets. Ground truth classification was determined by the gold standard Polymerase Chain Reaction (PCR) test results. A residual bootstrapped method generated the simulated datasets by fitting signal data to Autoregressive Moving Average (ARMA) models. Classification models included neural networks, k-nearest neighbors, naïve Bayes, random forest, and support vector machines. A Recursive Feature Elimination (RFE) study was performed to determine if reducing signal features would improve the classification models performance using Gini Importance scoring for the two classes. The top 25% of …

Evaluation Of Generative Models For Predicting Microstructure Geometries In Laser Powder Bed Fusion Additive Manufacturing, Andy Ramlatchan

Computer Science Theses & Dissertations

In-situ process monitoring for metals additive manufacturing is paramount to the successful build of an object for application in extreme or high stress environments. In selective laser melting additive manufacturing, the process by which a laser melts metal powder during the build will dictate the internal microstructure of that object once the metal cools and solidifies. The difficulty lies in that obtaining enough variety of data to quantify the internal microstructures for the evaluation of its physical properties is problematic, as the laser passes at high speeds over powder grains at a micrometer scale. Imaging the process in-situ is complex …

Utilization Of Finite Element Analysis Techniques For Adolescent Idiopathic Scoliosis Surgical Planning, Michael A. Polanco

Mechanical & Aerospace Engineering Theses & Dissertations

Adolescent Idiopathic Scoliosis, a three-dimensional deformity of the thoracolumbar spine, affects approximately 1-3% of patients ages 10-18. Surgical correction and treatment of the spinal column is a costly and high-risk task that is consistently complicated by factors such as patient-specific spinal deformities, curve flexibility, and surgeon experience. The following dissertation utilizes finite element analysis to develop a cost-effective, building-block approach by which surgical procedures and kinematic evaluations may be investigated. All studies conducted are based off a volumetric, thoracolumbar finite element (FE) model developed from computer-aided design (CAD) anatomy whose components are kinematically validated with in-vitro data. Spinal ligament stiffness …

Stability And Differential Privacy Of Stochastic Gradient Methods, Zhenhuan Yang

Legacy Theses & Dissertations (2009 - 2024)

Recently there are a considerable amount of work devoted to the study of the algorithmic stability as well as differential privacy (DP) for stochastic gradient methods (SGM). However, most of the existing work focus on the empirical risk minimization (ERM) and the population risk minimization problems. In this paper, we study two types of optimization problems that enjoy wide applications in modern machine learning, namely the minimax problem and the pairwise learning problem.

Symbolic Power Containments In Singular Rings In Positive Characteristic, Eloísa Grifo, Linquan Ma, Karl Schwede

Department of Mathematics: Faculty Publications

The containment problem for symbolic and ordinary powers of ideals asks for what values of a and b we have I(a)⊆Ib. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for F-purity (respectively, strong F-regularity) for ideals of finite projective dimension over F-finite Gorenstein rings and use …

Ocean Wave Prediction And Characterization For Intelligent Maritime Transportation, Pujan Pokhrel

LSU New Orleans Theses and Dissertations

The national Earth System Prediction (ESPC) initiative aims to develop the predictions
for the next generation predictions of atmosphere, ocean, and sea-ice interactions in the scale of days to decades. This dissertation seeks to demonstrate the methods we can use to improve the ESPC models, especially the ocean prediction model. In the application side of the weather forecasts, this dissertation explores imitation learning with constraints to solve combinatorial optimization problems, focusing on the weather routing of surface vessels. Prediction of ocean waves is essential for various purposes, including vessel routing, ocean energy harvesting, agriculture, etc. Since the machine learning approaches …

Cohomology Of The Symmetric Group With Twisted Coefficients And Quotients Of The Braid Group, Trevor Nakamura

Graduate Theses and Dissertations

In 2014 Brendle and Margalit proved the level $4$ congruence subgroup of the braid group, $B_{n}[4]$, is the subgroup of the pure braid group generated by squares of all elements, $PB_{n}^{2}$. We define the mod $4$ braid group, $\Z_{n}$, to be the quotient of the braid group by the level 4 congruence subgroup, $B_{n}/B_{n}[4]$. In this dissertation we construct a group presentation for $\Z_{n}$ and determine a normal generating set for $B_{n}[4]$ as a subgroup of the braid group. Further work by Kordek and Margalit in 2019 proved $\Z_{n}$ is an extension of the symmetric group, $S_{n}$, by $\mathbb{Z}_{2}^{\binom{n}{2}}$. A …

Automating Defeasible Reasoning In Law With Answer Set Programming, How Khang Lim, Avishkar Mahajar, Martin Strecker, Meng Weng Wong

Centre for Computational Law (2022-2025)

The paper studies defeasible reasoning in rule-based systems, in particular about legal norms and contracts. We identify rule modifiers that specify how rules interact and how they can be overridden. We then define rule transformations that eliminate these modifiers, leading in the end to a translation of rules to formulas. For reasoning with and about rules, we contrast two approaches, one in a classical logic with SMT solvers, which is only briefly sketched, and one using non-monotonic logic with Answer Set Programming solvers, described in more detail.

Human Perception Of Exponentially Increasing Data Displayed On A Log Scale Evaluated Through Experimental Graphics Tasks, Emily Robinson

Department of Statistics: Dissertations, Theses, and Student Research

Log scales are often used to display data over several orders of magnitude within one graph. We conducted a series of three graphical studies to evaluate the impact displaying data on the log scale has on human perception of exponentially increasing trends compared to displaying data on the linear scale. Each study was related to a different graphical task, each requiring a different level of interaction and cognitive use of the data being presented. The first experiment evaluated whether our ability to perceptually notice differences in exponentially increasing trends is impacted by the choice of scale. Participants were shown a …

Could Cultures Determine The Course Of Epidemics And Explain Waves Of Covid-19?, Md Salman Rahman

Theses and Dissertations

Coronavirus Disease (COVID-19), caused by the SARS-CoV-2 virus, is an infectious disease that quickly became a pandemic spreading with different patterns in each country. Travel bans, lockdowns, social distancing, and non-essential business closures caused significant economic disruptions and stalled growth worldwide in the pandemic’s first year. In almost every country, public health officials forced and/or encouraged Nonpharmaceutical Interventions (NPIs) such as contact tracing, social distancing, masks, and quarantine. Human behavioral decision-making regarding social isolation significantly impedes global success in containing the pandemic. This thesis focuses on human behaviors and cultures related to the decision-making of social isolation during the pandemic. …