Applied Mathematics Commons^™

Why 3d Fragmentation Usually Leads To Cuboids: A Simple Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been empirically observed that the average shape of natural fragmentation results -- such as natural rock fragments -- is a distorted cube (known as cuboid). Recently, a complex explanation was provides for this empirical fact. In this paper, we propose a simple geometry-based physical explanation for the ubiquity of cuboid fragments.

Two Runners In The Time Of Social Distancing, Speedboats In The Gulf Of Finland: How To Best Pass?, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If two runners follow the same running path, what is the best trajectory for the faster runner to pass the slower one, taking into account that they should always maintain a prescribed social distance? If a speedboat wants to pass a slower ship following a special canal in the Gulf of Finland, what is the best trajectory? In this paper, we provide answers to both questions.

The Similarity Between Earth's And Mars's Core-Mantle Boundary Seems To Be Statistically Significant, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Latest, most accurate measurements of the depth of the Mars's core-mantle boundary shows that the ratio between this depth and Mars's radius is the same as for the Earth -- and with new measurements, this coincidence has become statistically significance. This coincidence seems to confirm a simple scale-invariant model in which for planets of Earth-Mars type, this depth is proportional to the planet's radius. Of course, we need more observations to confirm this model, but the fact that, for the first time, we got a statistically significant confirmation, is encouraging: it makes us believe that this coincidence is not accidental.

Under Limited Resources, Lottery-Based Tutoring Is The Most Efficient, Olga Kosheleva, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, every student who needs tutoring should receive intensive one-on-one tutoring. In practical, schools' resources are limited, so the students get only a portion of needed tutoring. It would have been not so bad if, e.g., half-time tutoring would be half as efficient as the intensive one. However, research shows that half-time tutoring is, on average, 15 times less efficient -- and, e.g., for math tutoring 20 times less efficient. To increase the efficiency, we propose to randomly divide the students who need tutoring into equal-size groups, and each year (or each semester) provide intensive tutoring to …

Exploring Food Deserts And Environmental Impacts On Health In Chicago And Oregon, Sivasomasundari Arunarasu, Paulina Grzybowicz

altREU Projects

Food deserts are defined as, “an impoverished area where residents lack access to healthy foods”. This lack of access can be due to a combination of socioeconomic, geographic, and food-related variables, and has been proven to impact the health of residents in the area. In this project, several statistical and machine learning techniques are used to model the impact of food desserts and various other factors on health outcomes, including diabetes and obesity rates, in both the different neighborhoods in the City of Chicago and the various counties in the state of Oregon. The models are then used to determine …

A Novel Correction For The Adjusted Box-Pierce Test — New Risk Factors For Emergency Department Return Visits Within 72 Hours For Children With Respiratory Conditions — General Pediatric Model For Understanding And Predicting Prolonged Length Of Stay, Sidy Danioko

Computational and Data Sciences (PhD) Dissertations

This thesis represents the results of three research projects that underline the breadth and depth of my interests.

Firstly, I devoted some efforts to the well-known Box-Pierce goodness-of-fit tests for time series models which has been an important research topic over the last few decades. All previously proposed tests are focused on changes of the test statistics. Instead, I adopted a different approach that takes the best performing test and modifying the rejection region. Thus, I developed a semiparametric correction of the Adjusted Box-Pierce test that attains the best I error rates for all sample sizes and lags and outperforms …

Combating Covid On College Campuses: The Impact Of Structural Changes On Viral Transmissions, Jared Knofczynski, Aria Killebrew Bruehl, Ben Warner, Ryne Shelton

altREU Projects

One of the most significant issues in the COVID-19 pandemic is the reopening of schools while minimizing the transmission of coronavirus. Opportunities for evaluating the effectiveness of policies that might be utilized at such institutions are limited, as the necessary empirical data has not been gathered yet. Agent-based modeling, where various entities within an environment are simulated as agents, offers an opportunity to examine the effectiveness of various policies in a way that drastically minimizes the health and economic risks involved. Agent-based modeling is common within biology, ecology and other fields; and has seen some use within the coronavirus literature. …

Analyzing Network Topology For Ddos Mitigation Using The Abelian Sandpile Model, Bhavana Panchumarthi, Monroe Ame Stephenson

altREU Projects

A Distributed Denial of Service (DDoS) is a cyber attack, which is capable of triggering a cascading failure in the victim network. While DDoS attacks come in different forms, their general goal is to make a network's service unavailable to its users. A common, but risky, countermeasure is to blackhole or null route the source, or the attacked destination. When a server becomes a blackhole, or referred to as the sink in the paper, the data that is assigned to it "disappears" or gets deleted. Our research shows how mathematical modeling can propose an alternative blackholing strategy that could improve …

An Adaptive Approach To Gibbs’ Phenomenon, Jannatul Ferdous Chhoa

Master's Theses

Gibbs’ Phenomenon, an unusual behavior of functions with sharp jumps, is encountered while applying the Fourier Transform on them. The resulting reconstructions have high frequency oscillations near the jumps making the reconstructions far from being accurate. To get rid of the unwanted oscillations, we used the Lanczos sigma factor to adjust the Fourier series and we came across three cases. Out of the three, two of them failed to give us the right reconstructions because either it was removing the oscillations partially but not entirely or it was completely removing them but smoothing out the jumps a little too much. …

Linear Methods For Regression With Small Sample Sizes Relative To The Number Of Variables., Rajesh Sikder

Electronic Theses and Dissertations

In data sets where there are a small number of observations but a large number of variables observed for each observation, ordinary least squares estimation cannot be used for regression models. There are many alternative including stepwise regression, penalized methods such as ridge regression and the LASSO, and methods based on derived inputs such as principal components regression and partial least squares regression. In this thesis, these five methods are described. K-fold cross validation is also discussed as a way for determining regularization parameters for each method. The performance of these methods in estimation and prediction is also examined through …

Numerical Approximations Of Phase Field Equations With Physics Informed Neural Networks, Colby Wight

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Designing numerical algorithms for solving partial differential equations (PDEs) is one of the major research branches in applied and computational mathematics. Recently there has been some seminal work on solving PDEs using the deep neural networks. In particular, the Physics Informed Neural Network (PINN) has been shown to be effective in solving some classical partial differential equations. However, we find that this method is not sufficient in solving all types of equations and falls short in solving phase-field equations. In this thesis, we propose various techniques that add to the power of these networks. Mainly, we propose to embrace the …

Optimal Control Of Multiphase Free Boundary Problems For Nonlinear Parabolic Equations, Evan Cosgrove

Theses and Dissertations

Dissertation research is on the optimal control of systems with distributed parameters described by singular nonlinear partial differential equations (PDE) modeling multi-phase Stefan type second order parabolic free boundary problems. This type of free boundary problems arise in various applications, such as biomedical engineering problem on the laser ablation of biological tissues, aerospace engineering problem on the ice accretion in aircrafts mid-flight, biomedical problem on the growth of cancerous tumor, and many other phase transition processes in thermophysics and fluid mechanics. The aim of the optimal control of distributed free boundary systems is two fold: identification of functional parameters of …

Convolution Inequalities And Applications To Partial Differential Equations., Matthew Reynolds

Electronic Theses and Dissertations

In this dissertation we develop methods for obtaining the existence of mild solutions to certain partial differential equations with initial data in weighted L p spaces and apply them to some examples as well as improve the solutions to some known PDEs studied extensively in the literature. We begin by obtaining a version of a Stein-Weiss integral inequality which we will use to obtain general convolution inequalities in weighted L p spaces using the techniques of interpolation. We will then use these convolution inequalities to make estimates on PDEs that will help us obtain mild solutions as fixed points of …

Combination Of The Single-Valued Neutrosophic Fuzzy Set And The Soft Set With Applications In Decision-Making, Florentin Smarandache, Ahmed Mostafa Khalil, Dunqian Chao, A. A. Azzam, W. Alharby

Branch Mathematics and Statistics Faculty and Staff Publications

In this article, we propose a novel concept of the single-valued neutrosophic fuzzy soft set by combining the single-valued neutrosophic fuzzy set and the soft set. For possible applications, five kinds of operations (e.g., subset, equal, union, intersection, and complement) on single-valued neutrosophic fuzzy soft sets are presented. Then, several theoretical operations of single-valued neutrosophic fuzzy soft sets are given. In addition, the first type for the fuzzy decision-making based on single-valued neutrosophic fuzzy soft set matrix is constructed. Finally, we present the second type by using the AND operation of the single-valued neutrosophic fuzzy soft set for fuzzy decision-making …

Residual-Based Multi-Filter Methodology For All-Source Fault Detection, Exclusion, And Performance Monitoring, Juan Jurado, John F. Raquet, Christine M. Schubert Kabban Air Force Institute Of Technology, Jonathon S. Gipson

Faculty Publications

All-source navigation has become increasingly relevant over the past decade with the development of viable alternative sensor technologies. However, as the number and type of sensors informing a system increases, so does the probability of corrupting the system with sensor modeling errors, signal interference, and undetected faults. Though the latter of these has been extensively researched, the majority of existing approaches have constrained faults to biases and designed algorithms centered around the assumption of simultaneously redundant, synchronous sensors with valid measurement models, none of which are guaranteed for all-source systems. As part of an overall all-source assured or resilient navigation …

Complete Integrability And Discretization Of Euler Top And Manakov Top, Austin Marstaller

Theses and Dissertations

The Euler top is a completely integrable system with physical system implications and the Manakov top is its four-dimensional extension. We are concerned about their complete integrability and the preservation of this property under a specific discretization known as the Hirota-Kimura Discretization. Surprisingly, it is not guaranteed that under any discretization the conserved quantities are preserved and therefore they must be discovered. In this work we construct the Poisson bracket and Lax pair for each system and provide the Lie algebra background needed to do such such constructions.

Statistical Methods For Resolving Intratumor Heterogeneity With Single-Cell Dna Sequencing, Alexander Davis

Dissertations and Theses (Open Access)

Tumor cells have heterogeneous genotypes, which drives progression and treatment resistance. Such genetic intratumor heterogeneity plays a role in the process of clonal evolution that underlies tumor progression and treatment resistance. Single-cell DNA sequencing is a promising experimental method for studying intratumor heterogeneity, but brings unique statistical challenges in interpreting the resulting data. Researchers lack methods to determine whether sufficiently many cells have been sampled from a tumor. In addition, there are no proven computational methods for determining the ploidy of a cell, a necessary step in the determination of copy number. In this work, software for calculating probabilities from …

Through-The-Wall Radar Detection Using Machine Learning, Aihua W. Wood, Ryan Wood, Matthew Charnley

Faculty Publications

This paper explores the through-the-wall inverse scattering problem via machine learning. The reconstruction method seeks to discover the shape, location, and type of hidden objects behind walls, as well as identifying certain material properties of the targets. We simulate RF sources and receivers placed outside the room to generate observation data with objects randomly placed inside the room. We experiment with two types of neural networks and use an 80-20 train-test split for reconstruction and classification.

Numerical Simulation Of Low Reynolds Number Locomotion In Viscoelastic Media, Nesreen Abdulrahim Althobaiti

Theses and Dissertations

We use computational models to investigate 2D swimmers within various fluid media with low Reynolds Number. Extensions of the standard Immersed Boundary (IB) Method are proposed so that the fluid media may satisfy no slip, partial slip or free-slip conditions on the moving boundary. The fluid equations are solved through a Multigrid preconditioned GMRES solver. Our numerical results indicate that slip may lead to substantial speed enhancement for swimmers in a viscoelastic fluid, as well as in a viscoelastic two-fluid mixture. Under the slip conditions, the speed of locomotion is dependent in a nontrivial way on both the viscosity and …

Methods In Modeling Wildlife Disease From Model Selection To Parameterization With Multi-Scale Data, Ian Mcgahan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The effects of emerging wildlife diseases are global and profound, resulting in loss of human life, economic and agricultural impacts, declines in wildlife populations, and ecological disturbance. The spread of wildlife diseases can be viewed as the result of two simultaneous processes: spatial spread of wildlife populations and disease spread through a population. For many diseases these processes happen at different timescales, which is reflected in available data. These data come in two flavors: high-frequency, high-resolution telemetry data (e.g. GPS collar) and low-frequency, low-resolution presence-absence disease data. The multi-scale nature of these data makes analysis of such systems challenging. Mathematical …

Creative Assignments In Upper Level Undergraduate Courses Inspired By Mentoring Undergraduate Research Projects, Malgorzata A. Marciniak

Journal of Humanistic Mathematics

This article describes methods and approaches for incorporating creative projects in undergraduate mathematics courses for students of engineering and computer science in an urban community college. The topics and the grading rubrics of the projects go way beyond standard homework questions and contain elements of finding own project, incorporating historical background, inventing own questions and exercises, or demonstrating experiments to illustrate some aspects of the project. After analyzing challenges and outcomes of these projects, I identified several skills which help students be successful, including the skills of creativity. These skills are writing, oral presentation, math skills, and collaboration skills. I …

Lattice Of Maximal-Primary Ideals In Quadratic Orders, Ryan Bridges

Mathematics & Statistics ETDs

An order is a subring of the ring of integers of an algebraic extension, Peruginelli and Zanardo classified the lattices of orders with prime index inside te ring of integers of quadratic extensions of the rational numbers. The lattices are quite striking and have different layered structure depending on whether the prime is inert, split, or ramified. This thesis considers the orders which have prime power index inside the Gaussian integers. This is a nice generalization of the work of Peruginelli and Zanardo, and succeeds in a few classifications of specific instances of orders derived from inert primes.

Expected Resurgence Of Ideals Defining Gorenstein Rings, Eloísa Grifo, Craig Huneke, Vivek Mukundan

Department of Mathematics: Faculty Publications

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian.

"A Comparison Of Variable Selection Methods Using Bootstrap Samples From Environmental Metal Mixture Data", Paul-Yvann Djamen

Mathematics & Statistics ETDs

In this thesis, I studied a newly developed variable selection method SODA, and three customarily used variable selection methods: LASSO, Elastic net, and Random forest for environmental mixture data. The motivating datasets have neuro-developmental status as responses and metal measurements and demographic variables as covariates. The challenges for variable selections include (1) many measured metal concentrations are highly correlated, (2) there are many possible ways of modeling interactions among the metals, (3) the relationships between the outcomes and explanatory variables are possibly nonlinear, (4) the signal to noise ratio in the real data may be low. To compare these methods …

Quantitatively Motivated Model Development Framework: Downstream Analysis Effects Of Normalization Strategies, Jessica M. Rudd

Doctor of Data Science and Analytics Dissertations

Through a review of epistemological frameworks in social sciences, history of frameworks in statistics, as well as the current state of research, we establish that there appears to be no consistent, quantitatively motivated model development framework in data science, and the downstream analysis effects of various modeling choices are not uniformly documented. Examples are provided which illustrate that analytic choices, even if justifiable and statistically valid, have a downstream analysis effect on model results. This study proposes a unified model development framework that allows researchers to make statistically motivated modeling choices within the development pipeline. Additionally, a simulation study is …

Methods Of Uncertainty Quantification For Physical Parameters, Kellin Rumsey

Mathematics & Statistics ETDs

Uncertainty Quantification (UQ) is an umbrella term referring to a broad class of methods which typically involve the combination of computational modeling, experimental data and expert knowledge to study a physical system. A parameter, in the usual statistical sense, is said to be physical if it has a meaningful interpretation with respect to the physical system. Physical parameters can be viewed as inherent properties of a physical process and have a corresponding true value. Statistical inference for physical parameters is a challenging problem in UQ due to the inadequacy of the computer model. In this thesis, we provide a comprehensive …

Assessing The Validity Of Sentiment Analysis Measures Through Polychoric Correlation, Kelli N. Kasper

Mathematics & Statistics ETDs

Sentiment analysis methods extract the attitude of a text via systematic algorithms. To evaluate the validity of common sentiment analysis methods, we use polychoric correlation to compare computer-mediated methods and human-rated analogues. Our main topics of interest are the internal consistency of the raters' scores, the level of consensus among raters, and how well raters' scores correlate with those given by sentiment analysis methods for randomly collected Twitter data.

Our analysis found that there is good validity for methods that measure negative and positive sentiments in short texts, both in terms of inter-rater consistency and when comparing raters to computer-mediated …

An Investigation Of Gene Regulatory Network State Space Variability, Sara Faye Liesman

Theses and Dissertations

Genes are segments of DNA that provide a blueprint for cells and organisms to effectively control processes and regulations within individuals. There have been many attempts to quantify these processes, as a greater understanding of how genes operate could have large impacts on both personalized and precision medicine. Gene interactions are of particular interest, however, current biological methods can not easily reveal the details of these interactions. Therefore, we infer networks of interactions from gene expression data which we call a gene regulatory network, or GRN. Due to the robust behavior of genes and the inherent variability within interactions, models …

Projecting The Covid-19 Weekly Deaths And Hospitalizations For Jefferson County, Kentucky, Seyed Karimi, Natalie Dupre, W. Paul Mckinney, Bert B. Little, Naiya Patel, Sarah Moyer

The University of Louisville Journal of Respiratory Infections

Introduction: The trends in the numbers of active hospitalizations and fatalities caused by the COVID-19 in Jefferson County, Kentucky, were projected over the period May 7 to August 20, 2020.

Methods: The projections provided in this report are from a susceptible-exposed-infectious-recovered (SEIR) model. The model was calibrated using the COVID-19 transmission dynamics parameters from relevant literature and clinical dynamics parameters from the county’s data. The model was used to measure the impact of public health policy interventions designed to contain the infection. The policy was modeled by its intervention day and impact on the transmission of the virus such that …

An Improved Method For Spectroscopic Quality Classification, Elizabeth G. Mayer

Mathematics & Statistics ETDs

Spectral quality classification is a vital step in data cleaning before the

analysis of magnetic resonance spectroscopy (MRS) data can be done. This

analysis compares five methods of quality classification; three of these are

legacy methods, Maudsley et al. (2006), Zhang et al. (2018), and

Bustillo et al. (2020), and two newly created methods that used a random forests

classifier (RFC) to inform their classifications. We found that the random forest

classifier was the most accurate at predicting spectra quality (balanced

accuracy for RF of 88% vs legacy of 70%, 72%, or 72%). A

Random-Forests-Informed Filtering method (RFIFM) for quality …