Physical Sciences and Mathematics Commons^™

New Conforming Finite Elements Based On The De Rham Complexes For Some Fourth-Order Problems, Qian Zhang

Wayne State University Dissertations

In this dissertation, we discuss the conforming finite element discretization of high-order equations involving operators such as $(\curl\curl)^2$, $\grad\Delta\div$, and $-\curl\Delta\curl$. These operators appear in various models, such as continuum mechanics, inverse electromagnetic scattering theory, magnetohydrodynamics, and linear elasticity. Naively discretizing these operators and their corresponding eigenvalue problems using the existing $H^2$-conforming element would lead to spurious solutions in certain cases. Therefore, it is desirable to design conforming finite elements for equations containing these high-order differential operators.

The $\curl\curl$-conformity or $\grad\curl$-conformity requires that the tangential component of $\curl \bm u_h$ is continuous. Recall that the N\'ed\'elec element requires only the …

Aps March Meeting 2021 (Online) Updates On Scientific Research During Pandemic Times, Vianney Gimenez-Pinto

Title III Professional Development Reports

While the ongoing global pandemic continues to affect our everyday lives, researchers in Science, Technology, Engineering and Math found a way to come together at the American Physical Society (APS) March Meeting 2021. The conference was online-only and had more than 11,000 registered attendants who actively participated in the program during March 14- 19, 2021.

Design Project: Smart Headband, John Michel, Jack Durkin, Noah Lewis

Williams Honors College, Honors Research Projects

Concussion in sports is a prevalent medical issue. It can be difficult for medical professionals to diagnose concussions. With the fast pace nature of many sports, and the damaging effects of concussions, it is important that any concussion risks are assessed immediately. There is a growing trend of wearable technology that collects data such as steps and provides the wearer with in-depth information regarding their performance. The Smart Headband project created a wearable that can record impact data and provide the wearer with a detailed analysis on their risk of sustaining a concussion. The Smart Headband uses accelerometers and gyroscopes …

Obstructive Wiring Patterns To Circular Planarity In Electrical Networks, Hannah Lebo

Williams Honors College, Honors Research Projects

In order for an electrical network to be printed on a flat surface without changing the network’s input or output, it is important to consider if any wires will cross and if this problem can be avoided. If a circular network can be printed so that no wires cross, the network is said to be circular planar. In this paper, we identify a number of wiring patterns that make circular planarity impossible. We find exactly 3 wiring patterns using circular pairs with sets of two nodes, and we find exactly 78 wiring patterns using circular pairs with sets of three …

Buckling Loads Of A Graphene Layer Interacting With Rigid Substrates, Bradley Beckwith

Williams Honors College, Honors Research Projects

The goal of this project is to formulate a model that can predict the buckling of a graphene layer between two rigid substrates. The model will predict the buckling of the graphene layer when it is parallel to the substrates and an edge load is applied to the ends of the layer. Our main focus is to use the model to predict buckling loads given different assumptions for interaction forces between the graphene layer and the substrates. For this project continuum modeling will be used to create a model for the graphene buckling problem. This modeling leads to a total …

Application Of Optimal Control Theory To A Malaria Model, Cassidy Hill

Murray State Theses and Dissertations

With malaria still prevalent and considered to be one of the most devastating infectious diseases in the world, many scientific efforts have been made to reduce its impact. One such effort includes the construction of mathematical models. Mathematical models can be used to analyze malaria transmission dynamics in the human population. The development of these models allows researchers to consider the control measures necessary to reduce the prevalence of malaria infection and possibly eliminate it.

The model presented in this thesis will provide the relationship of female Anopheles mosquitoes and insecticide treated paint acting as the control. A deterministic system …

Modeling And Simulation Techniques Used In High Strain Rate Projectile Impact, Derek G. Spear, Anthony N. Palazotto, Ryan A. Kemnitz

Faculty Publications

A series of computational models and simulations were conducted for determining the dynamic responses of a solid metal projectile impacting a target under a prescribed high strain rate loading scenario in three-dimensional space. The focus of this study was placed on two different modeling techniques within finite element analysis available in the Abaqus software suite. The first analysis technique relied heavily on more traditional Lagrangian analysis methods utilizing a fixed mesh, while still taking advantage of the finite difference integration present under the explicit analysis approach. A symmetry reduced model using the Lagrangian coordinate system was also developed for comparison …

Extending Critical Infrastructure Element Longevity Using Constellation-Based Id Verification, Christopher M. Rondeau, Michael A. Temple, J. Addison Betances, Christine M. Schubert Kabban

Faculty Publications

This work supports a technical cradle-to-grave protection strategy aimed at extending the useful lifespan of Critical Infrastructure (CI) elements. This is done by improving mid-life operational protection measures through integration of reliable physical (PHY) layer security mechanisms. The goal is to improve existing protection that is heavily reliant on higher-layer mechanisms that are commonly targeted by cyberattack. Relative to prior device ID discrimination works, results herein reinforce the exploitability of constellation-based PHY layer features and the ability for those features to be practically implemented to enhance CI security. Prior work is extended by formalizing a device ID verification process that …

A Geometric Description Of The Set Of Stabilizing Pid Controllers, Keqin Gu, Qian Ma, Huiqing Zhou, Mahzoon Salma, Xingzi Yang

SIUE Faculty Research, Scholarship, and Creative Activity

This article developed a new method to described the set of stabilizing PID control. The method is based on D-parameterization with natural description of the set. It was found that the stability crossing surface is a ruled surface that is completely determined by a curve known as discriminant. The discriminant is divided into sectors at the cusps. Corresponding to the sectors, the stability crossing surface is divided into positive and negative patches. A systematic study is conducted to identify the regions with a fixed number of right half-plane characteristic roots. The crossing directions of characteristic roots for positive patches and …

A Gender And Race Theoretical And Probabilistic Analysis Of The Recent Title Ix Policy Changes, Jordan Wellington

Scripps Senior Theses

On May 6th, 2020, after extensive public comment and review, the Department of Education published the final rule for the new Title IX regulations, which took effect in schools on August 14th. Title IX is the nearly fifty year old piece of the Education Amendments that prohibits sexual discrimination in federally funded schools. Several of these changes, such as the inclusion of live hearings and cross examination of witnesses, have been widely criticized by victims’ rights advocates for potentially retraumatizing victims of sexual assault and discouraging students from pursuing a Title IX claim. While the impact of the new regulations …

Cross-Model Parameter Estimation In Epidemiology, Julia R. Fitzgibbons

Honors Theses and Capstones

No abstract provided.

Modeling Coupled Disease-Behavior Dynamics Of Sars-Cov-2 Using Influence Networks, Juliana C. Taube

Honors Projects

SARS-CoV-2, the virus that causes COVID-19, has caused significant human morbidity and mortality since its emergence in late 2019. Not only have over three million people died, but humans have been forced to change their behavior in a variety of ways, including limiting their contacts, social distancing, and wearing masks. Early infectious disease models, like the classical SIR model by Kermack and McKendrick, do not account for differing contact structures and behavior. More recent work has demonstrated that contact structures and behavior can considerably impact disease dynamics. We construct a coupled disease-behavior dynamical model for SARS-CoV-2 by incorporating heterogeneous contact …

Thermodynamic Entropy Of A Magnetized Ree-Eyring Particle-Fluid Motion With Irreversibility Process: A Mathematical Paradigm, M. M. Bhatti, Sara I. Abdelsalam

Basic Science Engineering

This article deals with the entropy generation and irreversibility process under the effects of partial slip on magnetic dusty liquid induced by peristaltic wave through a porous channel. The Ree-Eyring fluid model has been used for a governing flow. Mathematical modelling is based on Ohm's law, continuity equation, Darcy law and momentum equation. Analytical solutions are presented for fluid and particle phase. The effects of different pertinent parameters are considered for Newtonian and non-Newtonian cases. Numerical integration has been carried out using a computational software to analyse the pumping characteristics. The behaviour of velocity profile, trapping mechanism, entropy generation, Bejan …

Leveraging Elasticity To Uncover The Role Of Rabinowitsch Suspension Through A Wavelike Conduit: Consolidated Blood Suspension Application, Sara I. Abdelsalam, A. Z. Zaher

Basic Science Engineering

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along with heat transfer. It is of interest in this work to study the effects of EOFs, which are rigid spherical particles that are suspended in the Rabinowitsch fluid, the Grashof parameter, heat source, and elasticity on the shear stress of the Rabinowitsch fluid model and flow quantities. The solutions are achieved by taking long wavelength approximation with …

A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui

Honors Theses

In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under the symmetry group of $P$. With this measure, we prove …

New Results On Cyclic Compositions And Multicompositions, Silvana Ramaj

Electronic Theses and Dissertations

Integer compositions, cyclic compositions, and lately k-compositions, are an important topic in combinatorics and number theory. In this paper, we will explain, the general approach of using generating functions to study number sequences involving compositions, cyclic compositions, k-compositions, and the number of parts in each of them. After generating the data, some properties are observed and proved. Also, some interesting bijections involving Pell numbers and the Jacobsthal sequence are given.

Expanding Social Network Modeling Software And Agent Models For Diffusion Processes, Patrick Vaden Shepherd

Theses and Dissertations--Computer Science

In an increasingly digitally interconnected world, the study of social networks and their dynamics is burgeoning. Anthropologically, the ubiquity of online social networks has had striking implications for the condition of large portions of humanity. This technology has facilitated content creation of virtually all sorts, information sharing on an unprecedented scale, and connections and communities among people with similar interests and skills. The first part of my research is a social network evolution and visualization engine. Built on top of existing technologies, my software is designed to provide abstractions from the underlying libraries, drive real-time network evolution based on user-defined …

Untouchable Money And Impossible Clones: Applications Of Quantum Picturalism And Zx-Calculus, Shea A. Roccaforte

Senior Projects Spring 2021

Quantum Picturalism allows a new technique for researchers and students alike in the areas of quantum computation and quantum information. This picturalistic method represents fundamental math concepts and quantum theory in a diagrammatic manner. This method is a high-level language that allows for the exploitation of quantum weirdness. Using these techniques, quantum processes and the composition of those processes are highlighted as a structure referred to as process theory. Viewing these processes in a purely diagrammatic language allows for an unambiguous universal language for qubits, and the manipulation of these diagrams is referred to as ZX-calculus. These concepts allow for …

Gibbs Phenomenon For Jacobi Approximations, Riti Bahl

Senior Projects Spring 2021

The classical Gibbs phenomenon is a peculiarity that arises when approximating functions near a jump discontinuity with the Fourier series. Namely, the Fourier series "overshoots" (and "undershoots") the discontinuity by approximately 9% of the total jump. This same phenomenon, with the same value of the overshoot, has been shown to occur when approximating jump-discontinuous functions using specific families of orthogonal polynomials. In this paper, we extend these results and prove that the Gibbs phenomenon exists for approximations of functions with interior jump discontinuities with the two-parameter family of Jacobi polynomials P_n^(a,b)(x). In particular, we show that for …

Electronic Properties Of Flat And Curved Graphene Sheets, Deng Yanpei

Senior Projects Spring 2021

This paper explored the electronic properties of the graphene sheet and also developed basis for understanding the electronic properties of the curved graphene sheet. This paper began with setting up basic knowledge about solid-state physics including introducing band structure, band gap, crystal structure, and reviews for quantum mechanical operators. Then this paper described two potential models that are suitable for considering periodic potential: the weak potential and the tight-binding model. We discovered the tight-binding model is better for our graphene case and by applying this model we find the energies of the graphene sheet. Next, we constructed the 1D and …

Optimal Construction Of A Layer-Ordered Heap And Its Applications, Jake Pennington

Graduate Student Theses, Dissertations, & Professional Papers

The layer-ordered heap (LOH) is a simple data structure used in algorithms that perform optimal top-$k$ on $X+Y$, algorithms with the best known runtime for top-$k$ on $X_1+X_2+\cdots+X_m$, and the fastest method in practice for computing the most abundant isotopologue peaks in a chemical compound. In the analysis of these algorithms, the rank, $\alpha$, has been treated as a constant and $n$, the size of the array, has been treated as the sole parameter. Here, we explore the algorithmic complexity of LOH construction with $\alpha$ as a parameter, introduce a few algorithms for constructing LOHs, analyze their complexity in both …

A Learning Curve Model Accounting For The Flattening Effect In Production Cycles, Evan R. Boone, John J. Elshaw, Clay M. Koschnick, Jonathan D. Ritschel, Adedeji B. Badiru

Faculty Publications

We investigate production cost estimates to identify and model modifications to a prescribed learning curve. Our new model examines the learning rate as a decreasing function over time as opposed to a constant rate that is frequently used. The purpose of this research is to determine whether a new learning curve model could be implemented to reduce the error in cost estimates for production processes. A new model was created that mathematically allows for a “flattening effect,” which typically occurs later in the production process. This model was then compared to Wright’s learning curve, which is a popular method used …

Developing Natural Language Processing Instruments To Study Sociotechnical Systems, Thayer Alshaabi

Graduate College Dissertations and Theses

Identifying temporal linguistic patterns and tracing social amplification across communities has always been vital to understanding modern sociotechnical systems. Now, well into the age of information technology, the growing digitization of text archives powered by machine learning systems has enabled an enormous number of interdisciplinary studies to examine the coevolution of language and culture. However, most research in that domain investigates formal textual records, such as books and newspapers. In this work, I argue that the study of conversational text derived from social media is just as important. I present four case studies to identify and investigate societal developments in …

Proper Orthogonal Decomposition: New Approximation Theory And A New Computational Approach, Sarah Katherine Locke

Doctoral Dissertations

“Proper orthogonal decomposition (POD) projection errors and error bounds for POD reduced order models of partial differential equations have been studied by many. In this research we obtain new results regarding POD data approximation theory and present a new difference quotient (DQ) approach for computing the POD modes of the data.

First, we improve on earlier results concerning POD projection errors by extending to a more general framework that allows for non-orthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD data approximation errors, and also prove new pointwise convergence results and error bounds …

Scaling Up Exact Neural Network Compression By Relu Stability, Thiago Serra, Xin Yu, Abhinav Kumar, Srikumar Ramalingam

Faculty Conference Papers and Presentations

We can compress a rectifier network while exactly preserving its underlying functionality with respect to a given input domain if some of its neurons are stable. However, current approaches to determine the stability of neurons with Rectified Linear Unit (ReLU) activations require solving or finding a good approximation to multiple discrete optimization problems. In this work, we introduce an algorithm based on solving a single optimization problem to identify all stable neurons. Our approach is on median 183 times faster than the state-of-art method on CIFAR-10, which allows us to explore exact compression on deeper (5 x 100) and wider …

Using Twitter Api To Solve The Goat Debate: Michael Jordan Vs. Lebron James, Jordan Trey Leonard

CMC Senior Theses

Using a Twitter API, I gather and analyze tweets by performing sentiment analysis to solve the GOAT debate among professional athletes with the primary focus on comparing Michael Jordan and LeBron James. Athletes from the National Football League (NFL), the National Basketball Association (NBA), Major League Baseball (MLB), and the National Collegiate Athletic Association (NCAA) Division 1 Men's and Women's Basketball were selected to compare how sentiment polarity varies across sports. Sentiment polarity is measured by labeling text as "positive", "neutral", or "negative" which allows us to determine which athlete/sport is highly favored among the Twitter community when it comes …

A New Mathematical Theory For The Dynamics Of Large Tumor Populations, A Potential Mechanism For Cancer Dormancy & Recurrence And Experimental Observation Of Melanoma Progression In Zebrafish, Adeyinka A. Lesi

Dissertations and Theses

Cancer, a family of over a hundred disease varieties, results in 600,000 deaths in the U.S. alone. Yet, improvements in imaging technology to detect disease earlier, pharmaceutical developments to shrink or eliminate tumors, and modeling of biological interactions to guide treatment have prevented millions of deaths. Cancer patients with initially similar disease can experience vastly different outcomes, including sustained recovery, refractory disease or, remarkably, recurrence years after apparently successful treatment. The current understanding of such recurrences is that they depend on the random occurrence of critical mutations. Clearly, these biological changes appear to be sufficient for recurrence, but are they …

Deterministic And Statistical Methods For Inverse Problems With Partial Data, Yanfang Liu

Dissertations, Master's Theses and Master's Reports

Inverse problems with partial data have many applications in science and engineering. They are more challenging than the complete data cases since the lack of data increases ill-posedness and nonlinearity. The use of only deterministic or statistical methods might not provide satisfactory results. We propose to combine the deterministic and statistical methods to treat such inverse problems. The thesis is organized as follows.

In Chapter 1, we briefly introduce the inverse problems and their applications. The classical deterministic methods and Bayesian inversion are discussed. The chapter is concluded with a summary of contributions.

Chapter 2 considers the reconstruction of the …

Supercritical And Subcritical Pitchfork Bifurcations In A Buckling Problem For A Graphene Sheet Between Two Rigid Substrates, Jake Grdadolnik

Williams Honors College, Honors Research Projects

In this paper we study a model of the buckling of a sheet of graphene between two rigid substrates. We seek to understand the buckling of the sheet as the substrate separation is varied with a fixed load on each end of the sheet. We write down the expression for total energy of the system and from it derive a 2-point nonlinear boundary-value problem whose solutions are equilibrium configurations of the sheet. We cannot get an explicit solution. Instead, we perform a bifurcation analysis by using asymptotics to approximate solutions on the bifurcating branches near the bifurcation points. The bifurcating …

An Enumeration Of Nested Networks, Nathan Cornelius

Williams Honors College, Honors Research Projects

Nested networks have several applications in phylogenetics and electrical circuit theory. In many cases, there may exist more than one distinct network which correctly models a given data set. This proposes a combinatorial problem to determine all possible network solutions. In this paper, we partially solve this problem by developing exponential generating functions which enumerate all 1-nested and 2-nested unicyclic networks. We also describe our procedure to directly count all 1-nested and 2-nested networks and provide all 1-nested networks with 7, 8, and 9 terminal nodes.