Applied 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 …

A Unified Approach For Constructing Confidence Intervals And Hypothesis Tests Using H-Function, Weizhen Wang

Mathematics and Statistics Faculty Publications

We introduce a general method, named the h-function method, to unify the constructions of level-a exact test and 1-a exact confidence interval. Using this method, any confidence interval is improved as follows: i) an approximate interval, including a point estimator, is modified to an exact interval; ii) an exact interval is refined to be an interval that is a subset of the previous one. Two real datasets are used to illustrate the method.

Common Unit Conversions For Irrigation, Dean E. Eisenhauer, Derrel L. Martin, Derek M. Heeren, Glenn J. Hoffman

Open Learning Resources, General/Selected

A fact sheet that includes common unit conversions for irrigation in international system of units (SI) and United States customary system of units (USCS).

Includes units for: length, area, volume, flow, weight and mass, pressure, crop yield, temperature, and the United States public land survey system.

Measuring Machine Learning Model Uncertainty With Applications To Aerial Segmentation, Kevin James Cotton

CGU Theses & Dissertations

Machine learning model performance on both validation data and new data can be better measured and understood by leveraging uncertainty metrics at the time of prediction. These metrics can improve the model training process by indicating which training data need to be corrected and what part of the domain needs further annotation. The methods described have yet to reach mainstream adoption, and show great potential. Here, we survey the field of uncertainty metrics and provide a robust framework for its application to aerial segmentation. Uncertainty is divided into two types: aleatoric and epistemic. Aleatoric uncertainty arises from variations in training …

Basic Probability Theory, Jose Luis Menaldi

Mathematics Faculty Research Publications

Long title: Basic Probability Theory: Independent Random Variables and Sample Spaces. Chapters: Elementary Probability - Basic Probability - Canonical Sample Spaces - Working on Probability Spaces - A Solutions to Exercises.

Fixed-Point Proximity Minimization: A Theoretical Review And Numerical Study, Daniel Weddle, Jianfeng Guo

OUR Journal: ODU Undergraduate Research Journal

This study examines the relatively recent development of a “fixed-point proximity” approach to one type of minimization problem, considers its application to image denoising, and explores convergence and divergence of the iterative algorithm beyond a (previously supplied) theoretically guaranteed convergence bound on one of the parameters (𝜆). While reviewing the fixed-point proximity approach and its application to image denoising, we aim to communicate the concepts and details in a way that will facilitate understanding for undergraduates and for scholars from other subfields. In the latter portion of our study, the numerical experiment provides thought-provoking data on the effects that parameters …

Is Technological Progress A Random Walk? Examining Data From Space Travel, Michael Howell, Daniel Berleant, Hyacinthe Aboudja, Richard Segall, Peng-Hung Tsai

Journal of the Arkansas Academy of Science

Improvement in a variety of technologies can often be successful modeled using a general version of Moore’s law (i.e. exponential improvements over time). Another successful approach is Wright’s law, which models increases in technological capability as a function of an effort variable such as production. While these methods are useful, they do not provide prediction distributions, which would enable a better understanding of forecast quality

Farmer and Lafond (2016) developed a forecasting method which produces forecast distributions and is applicable to many kinds of technology. A fundamental assumption of their method is that technological progress can be modeled as a …

Fourth Down Decision Making: Challenging The Conservative Nature Of Nfl Coaches, Will Palmquist, Ryan Elmore, Benjamin Williams

DU Undergraduate Research Journal Archive

This thesis analyzes the hypothesis that coaches in the National Football League are often too conservative in their decision making on fourth downs. I used R Studio and NFL play-by-play data to simulate actual football plays and drives according to different fourth down strategies. By measuring expected points per drive over thousands of simulated drives, we are able to evaluate the effectiveness of different fourth down strategies. This research points to a number of conclusions regarding the nature of NFL coaches on fourth downs as well as the complexity of modeling and simulating decision making in a complex sport such …

Emergency Communications Deficiency Locator, Austin Collins

Undergraduate Journal of Mathematical Modeling: One + Two

The Bi-Directional Amplifier (BDA) is the newest edition to Life Safety in the state of Florida. The Florida Fire Prevention Code (NFPA 1) section 11.10.1 states that “In all new and existing buildings, minimum radio signal strength for fire department communications shall be maintained at a level determined by the authority having jurisdiction (Committee NFPA 1: Fire Code 2018). That authority having jurisdiction for our local Tampa Bay area is the Hillsborough County Fire Rescue department and they have posted their own requirements along with the Florida Senate for emergency communication standards. All existing “Hi-rise” buildings, 75 feet tall …

Coloring Permutation-Gain Graphs, Daniel Slilaty

Mathematics and Statistics Faculty Publications

Correspondence colorings of graphs were introduced in 2018by Dvoˇr ́ak and Postle as a generalization of list colorings of graphswhich generalizes ordinary graph coloring. Kim and Ozeki observed thatcorrespondence colorings generalize various notions of signed-graph col-orings which again generalizes ordinary graph colorings. In this notewe state how correspondence colorings generalize Zaslavsky’s notionof gain-graph colorings and then formulate a new coloring theory ofpermutation-gain graphs that sits between gain-graph coloring and cor-respondence colorings. Like Zaslavsky’s gain-graph coloring, our newnotion of coloring permutation-gain graphs has well defined chromaticpolynomials and lifts to colorings of the regular covering graph of apermutation-gain graph

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 …

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 …

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 …

Mathematical Modeling Of Lung Inflammation: Macrophage Polarization And Ventilator-Induced Lung Injury With Methods For Predicting Outcome, Sarah B. Minucci

Theses and Dissertations

Lung insults, such as respiratory infections and lung injuries, can damage the pulmonary epithelium, with the most severe cases needing mechanical ventilation for effective breathing and survival. Furthermore, despite the benefits of mechanical ventilators, prolonged or misuse of ventilators may lead to ventilation-associated/ventilation-induced lung injury (VILI). Damaged epithelial cells within the alveoli trigger a local immune response. A key immune cell is the macrophage, which can differentiate into a spectrum of phenotypes ranging from pro- to anti-inflammatory. To gain a greater understanding of the mechanisms of the immune response in the lungs and possible outcomes, we developed several mathematical models …

Solving And Applications Of Multi-Facility Location Problems, Anuj Bajaj

Wayne State University Dissertations

This thesis is devoted towards the study and solving of a new class of multi-facility location problems. This class is of a great theoretical interest both in variational analysis and optimization while being of high importance to a variety of practical applications. Optimization problems of this type cannot be reduced to convex programming like, the much more investigated facility location problems with only one center. In contrast, such classes of multi-facility location problems can be described by using DC (difference of convex) programming, which are significantly more involved from both theoretical and numerical viewpoints.In this thesis, we present a new …

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 …

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 …

Neither “Post-War” Nor Post-Pregnancy Paranoia: How America’S War On Drugs Continues To Perpetuate Disparate Incarceration Outcomes For Pregnant, Substance-Involved Offenders, Becca S. Zimmerman

Pitzer Senior Theses

This thesis investigates the unique interactions between pregnancy, substance involvement, and race as they relate to the War on Drugs and the hyper-incarceration of women. Using ordinary least square regression analyses and data from the Bureau of Justice Statistics’ 2016 Survey of Prison Inmates, I examine if (and how) pregnancy status, drug use, race, and their interactions influence two length of incarceration outcomes: sentence length and amount of time spent in jail between arrest and imprisonment. The results collectively indicate that pregnancy decreases length of incarceration outcomes for those offenders who are not substance-involved but not evenhandedly -- benefitting white …

Optimal Tile-Based Dna Self-Assembly Designs For Lattice Graphs And Platonic Solids, Leyda Almodovar, Joanna Ellis-Monaghan, Amanda Harsy, Cory Johnson, Jessica Sorrells

Mathematics Faculty Publications

A design goal in self-assembly of DNA nanostructures is to find minimal sets of branched junction molecules that will self-assemble into targeted structures. This process can be modeled using techniques from graph theory. This paper is a collection of proofs for a set of DNA complexes which can be represented by specific graphs, namely Platonic solids, square lattice graphs, and triangular lattice graphs. This work supplements the results presented in https://arxiv.org/abs/2108.00035

Some Proofs Regarding Minami Estimates And Local Eigenvalue Statistics For Some Random Schrödinger Operator Models, Samuel Herschenfeld

Theses and Dissertations--Mathematics

We provide three proofs on different, but related models in the field of random Schrödinger operators. All three results are motivated by the desire to extend results and techniques on eigenvalue statistics or Minami estimates (an essential ingredient Poisson eigenvalue statistics).

Chapters 2 and 4 are explorations of the only two known techniques for proving Minami estimates for continuum Minami estimates. In Chapter 2, we provide an alternative and simplified proof of Klopp that holds in d = 1. Chapter 4 is an application of the techniques of Dietlein and Elgart to prove a Minami estimate for finite rank lattice …

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 …

Multifractional Brownian Motion And Its Applications To Factor Analysis On Consumer Confidence Index, Christopher Box

CMC Senior Theses

This thesis aims at introducing a new way to model time series objects in statistics using multifractional processes. It provides a detailed review of Brownian motion, fractional Brownian motion and extends the above 2 models to multifractional processes. To demonstrate a successful application to the real world, we perform pattern analysis on consumer confidence and household spending behavior. The analysis is conducted through investigating the local Holder regularity of the consumer confidence index and household expenditure. In the analysis, we first model consumer confidence index and household expenditure with a multifractional stochastic processes. We then use the index, pointwise Holder …

Improving The Temporal Accuracy Of Turbulence Models And Resolving The Implementation Issues Of Fluid Flow Modeling, Kyle J. Schwiebert

Dissertations, Master's Theses and Master's Reports

A sizeable proportion of the work in this thesis focuses on a new turbulence model, dubbed ADC (the approximate deconvolution model with defect correction). The ADC is improved upon using spectral deferred correction, a means of constructing a higher order ODE solver. Since both the ADC and SDC are based on a predictor-corrector approach, SDC is incorporated with essentially no additional computational cost. We will show theoretically and using numerical tests that the new scheme is indeed higher order in time than the original, and that the benefits of defect correction, on which the ADC is based, are preserved.

The …

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 …

Five-Wave Resonances In Deep Water Gravity Waves: Integrability, Numerical Simulations And Experiments, Dan Lucas, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, Miguel D. Bustamante

Articles

In this work we consider the problem of finding the simplest arrangement of resonant deep water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wave vectors K₁ + K₂ = K₃ (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wave packets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction …

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 …

A Mathematical Analysis Of The Wind Triangle Problem And An Inquiry Of True Airspeed Calculations In Supersonic Flight, Leonard T. Huang, Lisa I. Cummings

International Journal of Aviation, Aeronautics, and Aerospace

In the first half of this paper, we present a fresh perspective toward the Wind Triangle Problem in aerial navigation by deriving necessary and sufficient conditions, which we call "go/no-go conditions", for the existence/non-existence of a solution of the problem. Although our derivation is based on simple trigonometry and basic properties of quadratic functions, it is mathematically rigorous. We also offer examples to demonstrate how easy it is to check these conditions graphically. In the second half of this paper, we use function theory to re-examine another problem in aerial navigation, namely, that of computing true airspeed — even in …

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 …

Hands-On Developmental Playbook For Stem Clubs In Elementary Schools, James Lipuma, Bruce G. Bukiet, Cristo Leon

STEM for Success Resources

The following guide is the result of the work of NJIT researchers, staff, and student STEM role models along with the many public and private partners including Apple and the Us ARMY. The work was funded by the National Science Foundation through the NSF INCLUDES Life grant and support from Howmet Aerospace Foundation grants. The materials in the guide have been developed and tested with our New Jersey school district partners in Hillside, Long Branch, Morris Plains, Newark, Princeton, and Weehawken. See our strategic partners section for a more complete list of partners and collaborators.

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 …