Applied Mathematics Commons^™

Discovering Kepler’S Third Law From Planetary Data, Boyan Kostadinov, Satyanand Singh

Publications and Research

In this data-inspired project, we illustrate how Kepler’s Third Law of Planetary Motion can be discovered from fitting a power model to real planetary data obtained from NASA, using regression modeling. The power model can be linearized, thus we can use linear regression to fit the model parameters to the data, but we also show how a non-linear regression can be implemented, using the R programming language. Our work also illustrates how the linear least squares used for fitting the power model can be implemented in Desmos, which could serve as the computational foundation for this project at a lower …

Optimal Control Of Algae Biofilm Growth In Wastewater Treatment Using Computational Mathematical Models, Gerald Benjamin Jones

Undergraduate Honors Capstone Projects

Microalgal biofilms are comprised of a syntrophic consortium of microalgae and other microorganisms embedded within an extracellular matrix. Despite significant processes in the application of microalgal biofilms in wastewater treatment, mechanistic understanding and optimization of microalgal biomass yield and productivity under environmental constraints is still lacking. This paper identifies theoretical insights on this challenging biological problem by leveraging novel mathematical and computational tools. In particular, through a computational mathematical model to advance the understanding of microalgal biofilm growth kinetics under environmental constraints through a systematic parameter study. Moreover, design of algae biofilm reactors for optimal biomass yield and productivity in …

Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

Electronic Theses and Dissertations

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …

Predicting Suboptimal Care In Insured Nebraskans With Known And Suspected Chronic Conditions., Avery Wallace

Capstone Experience: Master of Public Health

Health insurance companies have a goal of improving population health for their members (people who the company insures). As a health insurance company, [Company] has ample data on the health of its members that can be utilized to improve the health, and by extension lives of the people they insure. Although [Company] does not deliver care, they communicate with their members and physicians to identify ways to improve the health of the member. Of specific interest are members with known chronic conditions who are receiving suboptimal care, as well as members who have undiagnosed chronic conditions. Two chronic conditions were …

The Brezis-Nirenberg Problem For The Generalized Kirchhoff Equation, Erisa Hasani

Theses and Dissertations

We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.

Parallel Computation Of Action Potentials In The Hodgkin-Huxley Model Via The Parareal Algorithm, Eric Boerman, Khanh Pham, Katie Peltier

Mathematics Student Work

The Hodgkin-Huxley model is a system of differential equations that describe the membrane voltage of an axon as it fires the basic signal of the nervous system: the action potential. When charge-carrying ions such as sodium, potassium, and others are enabled to cross a selectively permeable membrane, the resulting current propagates along the length of the axon as a wave of altered ionic potential. However, the degree to which the membrane is permeable to sodium and potassium is itself gated by voltage; therefore, voltage depends on permeability and permeability depends on voltage. This interdependent cellular system is expressed as a …

Solving The Heat Equation With Interfaces, Michael Bauer, Rex Llewellyn, Shauna Frank

Mathematics Student Work

When modeling systems made up of two materials with different thermodynamic properties, a physical interface can be introduced to account for the border where the materials meet. This interface separates our model’s standard grid into two regions, each with its unique physical properties. At these interfaces, boundary conditions can be imposed to represent the difference in heat and in heat flux between the different materials so that their interaction may be modeled accurately. Because standard finite difference methods are inadequate to deal with interfaces, a Matched Interface and Boundary (MIB) technique is investigated in this work to solve the heat …

Analysis Of An Integral Metric On Hyperspaces, Darren Schmidt

Undergraduate Research Conference at Missouri S&T

In this paper, we will be investigating how to compute the integral distance defined by Dr. Insall and Dr. Charatonik, and we will analyze the results from this computation. We develop a way to compute the integral distance by using Monte-Carlo Integration, and we analyze the time complexity and the error that results from this method of computation. We also investigate when this distance function is a metric, and how this metric compares to some other common metrics.

Construct Linear Quasi-Interpolants On Infinite Intervals, Johara Farah Albaliwi

Dissertations

In solving the data interpolation problem, which is fundamental in data analysis, we typically deal with the data samples spread in a finite interval [a, b], which results in the operations involving finite-dimensional matrices. There are many interesting results developed under this framework. However, when the data samples are given from an infinite interval [a, ∞) (for certain special types of real-world applications), many existing results would not work anymore due to the special properties of the infinite data samples. A new framework should be established to support the infinite data samples.

In this dissertation, we develop a special tool …

Mathematics And Enterprise Innovation, Pingwen Zhang

Bulletin of Chinese Academy of Sciences (Chinese Version)

The innovation and development of China are inseparable from mathematics. The development of applied mathematics, embodied in scientific discovery, national defense construction and enterprise innovation, is mainly driven by national demand. At present, China's economy has entered into a period of innovation driven development. Enterprises, as the main participants of national economic activities, need the support of mathematics for innovation and development. Regarding how to promote enterprise innovation through mathematics, this paper puts forward four aspects that we need to pay attention to and improve on: posing problems, solving problems, reporting results, and evaluating results. At the end, the paper …

Compact Dupin Hypersurfaces, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface M in Rⁿ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on M, i.e., each continuous principal curvature function has constant multiplicity on M. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in Rⁿ or Sⁿ . The theory of compact proper Dupin hypersurfaces in Sⁿ is closely related to the theory of isoparametric hypersurfaces in S^{n …}

Lecture 08: Partial Eigen Decomposition Of Large Symmetric Matrices Via Thick-Restart Lanczos With Explicit External Deflation And Its Communication-Avoiding Variant, Zhaojun Bai

Mathematical Sciences Spring Lecture Series

There are continual and compelling needs for computing many eigenpairs of very large Hermitian matrix in physical simulations and data analysis. Though the Lanczos method is effective for computing a few eigenvalues, it can be expensive for computing a large number of eigenvalues. To improve the performance of the Lanczos method, in this talk, we will present a combination of explicit external deflation (EED) with an s-step variant of thick-restart Lanczos (s-step TRLan). The s-step Lanczos method can achieve an order of s reduction in data movement while the EED enables to compute eigenpairs in batches along with a number …

Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai

Mathematical Sciences Spring Lecture Series

We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton's method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, …

Predicting Tumor Response To Radiotherapy Based On Estimation Of Non-Treatment Parameters, Yutian Huang, Allison L. Lewis

Spora: A Journal of Biomathematics

Though clinicians can now collect detailed information about a variety of tumor characteristics as a tumor evolves, it remains difficult to predict the efficacy of a given treatment prior to administration. Additionally, the process of data collection may be invasive and expensive. Thus, the creation of a framework for predicting patient response to treatment using only information collected prior to the start of treatment could be invaluable. In this study, we employ ordinary differential equation models for tumor growth and utilize synthetic data from a cellular automaton model for calibration. We investigate which parameters have the most influence upon treatment …

Kleptoparasitic Hawk-Dove Games, Isabella H. Evans-Riester, Chasity T. Kay, Karina L. Ortiz-Suarez, Jan Rychtář, Dewey Taylor

Spora: A Journal of Biomathematics

The Hawk-Dove game is a classical game-theoretical model of potentially aggressive animal conflicts. In this paper, we apply game theory to a population of foraging animals that may engage in stealing food from one another. We assume that the population is composed of two types of individuals, Hawks and Doves. Hawks try to escalate encounters into aggressive contests while Doves engage in non-aggressive displays between themselves or concede to aggressive Hawks. The fitness of each type depends upon various natural parameters, such as food density, the mean handling time of a food item, as well as the mean times of …

Netsci High: Bringing Agency To Diverse Teens Through The Science Of Connected Systems, Stephen M. Uzzo, Catherine B. Cramer, Hiroki Sayama, Russell Faux

Northeast Journal of Complex Systems (NEJCS)

This paper follows NetSci High, a decade-long initiative to inspire teams of teenage researchers to develop, execute and disseminate original research in network science. The project introduced high school students to the computer-based analysis of networks, and instilled in the participants the habits of mind to deepen inquiry in connected systems and statistics, and to sustain interest in continuing to study and pursue careers in fields involving network analysis. Goals of NetSci High ranged from proximal learning outcomes (e.g., increasing high school student competencies in computing and improving student attitudes toward computing) to highly distal (e.g., preparing students for 21st …

Lecture 10: Preconditioned Iterative Methods For Linear Systems, Edmond Chow

Mathematical Sciences Spring Lecture Series

Iterative methods for the solution of linear systems of equations – such as stationary, semi-iterative, and Krylov subspace methods – are classical methods taught in numerical analysis courses, but adapting these methods to run efficiently at large-scale on high-performance computers is challenging and a constantly evolving topic. Preconditioners – necessary to aid the convergence of iterative methods – come in many forms, from algebraic to physics-based, are regularly being developed for linear systems from different classes of problems, and similarly are evolving with high-performance computers. This lecture will cover the background and some recent developments on iterative methods and preconditioning …

Lecture 00: Opening Remarks: 46th Spring Lecture Series, Tulin Kaman

Mathematical Sciences Spring Lecture Series

Opening remarks for the 46th Annual Mathematical Sciences Spring Lecture Series at the University of Arkansas, Fayetteville.

A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Gamage, Yan Peng

College of Sciences Posters

Interface problems have many applications in fluid dynamics, molecular biology, electromagnetism, material science, heat distribution in engines, and hyperthermia treatment of cancer. Mathematically, interface problems commonly lead to partial differential equations (PDE) whose in- put data are discontinuous or singular across the interfaces in the solution domain. Many standard numerical methods designed for smooth solutions poorly work for interface problems as solutions of the interface problems are mostly non-smoothness or discontinuous. Moving interface problems depends on the accuracy of the gradient of the solution at the interface. Therefore, it became essential to derive a method for interface problems that gives …

End-To-End Physics Event Generator, Yasir Alanazi, N. Sato, Tianbo Liu, W. Melnitchouk, Michelle P. Kuchera, Evan Pritchard, Michael Robertson, Ryan Strauss, Luisa Velasco, Yaohang Li

College of Sciences Posters

We apply generative adversarial network (GAN) technology to build an event generator that simulates particle production in electron-proton scattering that is free of theoretical assumptions about underlying particle dynamics. The difficulty of efficiently training a GAN event simulator lies in learning the complicated pat- terns of the distributions of the particles physical properties. We develop a GAN that selects a set of transformed features from particle momenta that can be generated easily by the generator, and uses these to produce a set of augmented features that improve the sensitivity of the discriminator. The new Feature-Augmented and Transformed GAN (FAT-GAN) is …

Instantaneous Frequency Estimation And Signal Separation Using Fractional Continuous Wavelet Transform, Abdelbaset R. Zeyani

Dissertations

In the signal processing field, time-frequency representations (TFR's) have intensively been improved to provide effective and powerful tools for reliable signal analysis. One of the most valuable and frequently used tools is Fourier transform (FT) which has been used to study the frequency content of stationary signals in the Fourier domain (FD). However, FT is not sufficient to study the frequency of non-stationary signals. For this particular type of signals to be best analyzed, some transforms such as the short time Fourier transform (STFT) and the continuous wavelet transform (CWT) have been introduced to provide us with a signal representation …

Analysis Of Boundary Observability Of Strongly Coupled One-Dimensional Wave Equations With Mixed Boundary Conditions, Wilson Dennis Horner

Masters Theses & Specialist Projects

*see note below

In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, speed of the wave propagation, etc. Suppose we have a strongly coupled system of two wave equations describing the longitudinal vibrations on a piezoelectric beam of length L. These two wave equations have non-identical wave propagation speeds c1 and c2. First, we prove the exact observability inequality with the optimal …

Modeling The Stock Market Through Game Theory, Kylie Hannafey

Honors College Theses

Game Theory is used on many occasions to help us understand interactions between decision-makers. The famous Nash equilibrium is a steady state in a model that shows the interaction of different players, in which no player can do better by choosing a different action if the actions of the other players do not change. These two concepts can be applied to numerous situations that vary in types of players, but for our research, we are focusing on businesses in the stock market. The main objective is to use Game Theory to analyze data collected from the stock market, model our …

Modeling Covid-19 Infection Rates Using Sir And Arima Models, Janelle Domantay, Ilya Pivavaruk, Victor Taksheyev

Undergraduate Research Symposium Posters

With the onset of the COVID-19 pandemic, it has become of increasing interest to both monitor and predict the growth of its infection rates. In order to analyze the accuracy of epidemiological prediction, we consider two different models for prediction, the Susceptible Infected and Removed (SIR), and Autoregressive Integrated Moving Average (ARIMA) models. Using a dataset of Clark County COVID-19 infections, we create various ARIMA and SIR models that attempt to predict the progression of COVID-19 infections whilst comparing these predictions to the dataset. We observed that the ARIMA model performed more accurately overall, having a much lower Root Mean …

The Beauty Of Bézier Curves, Qing Chen, Ariane Masuda

Publications and Research

It is very difficult for ordinary people to become excellent painters like Picasso. In contemporary society, everyone has a computer, but no one associates painting with computers. This project aims to show that one can use computer tools to connect mathematics with art. We use Krita, which is a professional free, and open-source painting program made by artists to create digital art. We demonstrate how the Bezier curve pen tool in Krita can help anyone to ́ draw paintings such as Picasso’s cubist oil paintings on a computer in a relatively short time.

The Fundamental Limit Theorem Of Countable Markov Chains, Nathanael Gentry

Senior Honors Theses

In 1906, the Russian probabilist A.A. Markov proved that the independence of a sequence of random variables is not a necessary condition for a law of large numbers to exist on that sequence. Markov's sequences -- today known as Markov chains -- touch several deep results in dynamical systems theory and have found wide application in bibliometrics, linguistics, artificial intelligence, and statistical mechanics. After developing the appropriate background, we prove a modern formulation of the law of large numbers (fundamental theorem) for simple countable Markov chains and develop an elementary notion of ergodicity. Then, we apply these chain convergence results …

Bézier Curves, Qing Chen, Ariane Masuda

Publications and Research

Drawing on a computer using a mouse is quite different than drawing by hand. It can be challenging to use a mouse to even simply trace a line. If the drawing involves several lines and curves, the task becomes more complicated. The goal of this project is to show how to design beautiful artworks using Bézier curves. A Bézier curve is a smooth parametric curve produced by the coordinates of certain points. To draw a specific curve, one needs to select multiple control points positioned in strategic places. By changing these positions, one can draw different curves to produce the …

0-Concordance Of 2-Knots, Nathan Sunukjian

University Faculty Publications and Creative Works

In this paper we investigate the 0-concordance classes of 2-knots in S4, an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin’s invariant, and invariants arising from Heegaard–Floer homology, we will prove that there are infinitely many 0-concordance classes of 2-knots.

Lie Groups And Euler-Bernoulli Beam Equation, Medeu Amangeldi

Theses and Dissertations

Lie groups approach in differential equations was a breakthrough subject in the late nineteenth century. Sophus Lie, a Norwegian mathematician, introduced the systematic approach to study the solutions of differential equations. The main goal of this thesis is to study, using Lie's approach, the Euler-Bernoulli beam equation subject to swelling force, the fourth-order nonlinear differential equation used to describe the beam deflection under the swelling force. In particular, we will classify the symmetry groups of this equation, obtain several reductions, and demonstrate both analytical and numerical solutions.

Flocc: From Agent-Based Models To Interactive Simulations On The Web, Scott Donaldson

Northeast Journal of Complex Systems (NEJCS)

Agent-based modeling (ABM) is a computational technique wherein systems are represented through the actions and interactions of many individual entities (‘agents’) over time. ABM often attempts to elucidate the unpredictable, high-level behavior of systems through the predictable, low-level behavior of actors within the system. There are currently few software or frameworks for ABM that allow modelers to design and build interactive models on the web, for a wide audience as well as a scientifically literate audience well-versed in complexity, models, and simulations. Flocc is a novel framework for agent-based modeling written in JavaScript, the lingua franca programming language of the …