Mathematics Commons^™

Classification Of Cayley Rose Window Graphs, Angsuman Das, Arnab Mandal

Theory & Applications of Graphs

Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs.

K-Classes Of Brill–Noether Loci And A Determinantal Formula, D. Anderson, Linda Chen, N. Tarasca

Mathematics & Statistics Faculty Works

We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number ρ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when ρ=1⁠, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for …

An Exploratory Analysis Of The Bgsu Learning Commons Student Usage Data, Emily Eskuri

Honors Projects

The purpose of this study was to explore past student usage data in individualized tutoring sessions from the Learning Commons from two academic years. The Bowling Green State University (BGSU) Learning Commons is a learning assistance center that offers various services, such as individualized tutoring, math assistance, writing assistance, study hours, and academic coaching. There have been limited research studies into how big data and analytics can have an impact in higher education, especially research utilizing predictive analytics.

This project applied analytics to individualized tutoring data in the Learning Commons to create a better understanding of why those trends happen …

Social Distancing And Testing As Optimal Strategies Against The Spread Of Covid-19 In The Rio Grande Valley Of Texas, Kristina P. Vatcheva, Josef A. Sifuentes, Tamer Oraby, Jose E. Campo Maldonado, Timothy Huber, Cristina Villalobos

School of Mathematical & Statistical Sciences Faculty Publications

At the beginning of August 2020, the Rio Grande Valley (RGV) of Texas experienced a rapid increase of coronavirus disease 2019 (abbreviated as COVID-19) cases and deaths. This study aims to determine the optimal levels of effective social distancing and testing to slow the virus spread at the outset of the pandemic. We use an age-stratified eight compartment epidemiological model to depict COVID-19 transmission in the community and within households. With a simulated 120-day outbreak period data we obtain a post 180-days period optimal control strategy solution. Our results show that easing social distancing between adults by the end of …

Developing Better Instruction, Better Instructors, And New Investigators, Clay Vander Kolk, Abigail Pyle

Celebration of Scholarship 2021

This research discusses how HAMTE and future educators collaborated to use the Principles to Action Professional Learning Toolkit in their classes and the tools they used to measure and gather conclusions about the development in the future teachers.

Normality Properties Of Composition Operators, Grace Weeks, Hallie Kaiser, Katy O'Malley

Celebration of Scholarship 2021

We explore two main concepts in relation to truncated composition matrices: the conditions required for the binormal and commutative properties. Both of these topics are important in linear algebra due to their connection with diagonalization.

We begin with the normal solution before moving onto the more complex binormal solutions. Then we cover conditions for the composition matrix to commute with the general matrix. Finally, we end with ongoing questions for future work.

Meta-Heuristic Optimization Methods For Quaternion-Valued Neural Networks, Jeremiah Bill, Lance E. Champagne, Bruce Cox, Trevor J. Bihl

Faculty Publications

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued …

Positive Solutions For A Fractional Boundary Value Problem With Lidstone Like Boundary Conditions, Jeffrey T. Neugebauer, Aaron G. Wingo

EKU Faculty and Staff Scholarship

We consider a higher order fractional boundary value problem with Lidstone like boundary conditions, where the nonlinearity is an L¹-Carathèodory function. We first consider the lower order problem. Then, by using a convolution to construct the Green’s function for the higher order problem, we are able to apply a recent fixed point theorem to show the existence of positive solutions of the boundary value problem.

An Agent-Based Model To Evaluate The Effect Of Socioeconomic Status And Demographic Factors On Covid-19 Prevalence And Mortality, Jonathan Huang, Berke Nouri

Arts & Sciences Student Symposium

The purpose of the project is to discover how the prevalence and mortality of a pandemic change depending on a population’s demographic factors as well as various intervention policies through a NetLogo agent-based model. This model will simulate how demographic factors affect the course of COVID-19 (infection rate, recovery rate, and death rate). Demographic factors of interest will include population density, income distribution, age distribution, and number of hospital beds per capita. Intervention policies include vaccination, social distancing, mask wearing, mass testing, and quarantining. Conclusions about the effect of demographic factors on the infection, recovery, and death rate will be …

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 …

On Tests Of Independence Among Multiplerandom Vectors Of Arbitrary Dimensions., Angshuman Roy Dr.

Doctoral Theses

Measures of dependence among several random vectors and associated tests of independence play a major role in different statistical applications. Blind source separation or independent component analysis (see, e.g., Hyv¨arinen et al., 2001; Shen et al., 2009), feature selection and feature extraction (see, e.g., Li et al., 2012), detection of serial correlation in time series (see, e.g., Ghoudi et al., 2001) and finding the causal relationships among the variables (see, e.g., Chakraborty and Zhang, 2019) are some examples of their wide-spread applications. Tests of independence has vast applications in other areas of sciences as well. For instance, to characterize the …

Improved Functional Outcome After Peripheral Nerve Stimulation Of The Impaired Forelimb Post-Stroke, Shih-Yen Tsai, Jennifer A. Schreiber, Natalie S. Adamczyk, Joanna Y. Wu, Son T. Ton, Ryan C. Hofler, James S. Walter, Timothy O'Brien, Gwendolyn Kartje-Tillotson, Russ P. Nockels

Mathematics and Statistics: Faculty Publications and Other Works

Lack of blood flow to the brain, i.e., ischemic stroke, results in loss of nerve cells and therefore loss of function in the effected brain regions. There is no effective treatment to improve lost function except restoring blood flow within the first several hours. Rehabilitation strategies are widely used with limited success. The purpose of this study was to examine the effect of electrical stimulation on the impaired upper extremity to improve functional recovery after stroke. We developed a rodent model using an electrode cuff implant onto a single peripheral nerve (median nerve) of the paretic forelimb and applied daily …

A Single-Scale Fractal Feature For Classification Of Color Images: A Virus Case Study, Walker Arce, James E. Pierce, Mihaela T. Velcsov

Mathematics Faculty Publications

Current methods of fractal analysis rely on capturing approximations of an images’ fractal dimension by distributing iteratively smaller boxes over the image, counting the set of box and fractal, and using linear regression estimators to estimate the slope of the set count line. To minimize the estimation error in those methods, our aim in this study was to derive a generalized fractal feature that operates without iterative box sizes or any linear regression estimators. To do this, we adapted the Minkowski-Bouligand box counting dimension to a generalized form by fixing the box size to the smallest fundamental unit (the individual …

New Characterizations Of Reproducing Kernel Hilbert Spaces And Applications To Metric Geometry, Daniel Alpay, Palle E. T. Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

Factoring: Difference Of Squares, Thomas Lauria

Open Educational Resources

This lesson plan will explain how to factor basic difference of squares problems

Matrix Product Structure Of A Permuted Quasi Cyclic Code And Its Dual, Perian Perdhiku

Dissertations

In my Dissertation I will work mostly with Permuted Quasi Cyclic Codes. They are a generalization of Cyclic Codes, one of the most important families of Linear Codes in Coding Theory. Linear Codes are very useful in error detection and correction. Error Detection and Correction is a technique that first detects the corrupted data sent from some transmitter over unreliable communication channels and then corrects the errors and reconstructs the original data. Unlike linear codes, cyclic codes are used to correct errors where the pattern is not clear and the error occurs in a short segment of …

Preliminary Results On The Use Of Machine Learning For Advance Controls Of A Dielectric Elastomer Actuator, Nathanael Gentry, Carson Farmer, Hector Medina

Liberty University Research Week

Undergraduate

Applied

Potential Connections Between Graph Theory And Materials Modeling, Rachel Drake, Hector Medina

Liberty University Research Week

Undergraduate

Applied

Ready To Engage? Urban Middle School Teachers’ Responsiveness To Targeted Engagement Interventions On Their Virtual Instructional Practices: An Action Research Study, Svetlana Nikic

Dissertations

Teachers’ effectiveness is associated with their instructional practices and is ultimately linked to students’ learning outcomes. In order to impact teachers’ effectiveness, schools focus substantial effort and resources on professional development led by an assumption that teachers’ classroom practices can be improved through targeted interventions. Even if this premise is correct, little information is available about how much a teacher’s practice may change through interventions, or which aspects of instructional practice are more receptive to improving teacher effectiveness (Garret et al., 2019).

This study took place at an urban middle school and examined teachers’ responsiveness to targeted engagement intervention in …

Understanding The Effect Of Adaptive Mutations On The Three-Dimensional Structure Of Rna, Justin Cook

Undergraduate Research and Scholarship Symposium

Single-nucleotide polymorphisms (SNPs) are variations in the genome where one base pair can differ between individuals.¹ SNPs occur throughout the genome and can correlate to a disease-state if they occur in a functional region of DNA.¹According to the central dogma of molecular biology, any variation in the DNA sequence will have a direct effect on the RNA sequence and will potentially alter the identity or conformation of a protein product. A single RNA molecule, due to intramolecular base pairing, can acquire a plethora of 3-D conformations that are described by its structural ensemble. One SNP, rs12477830, which …

Translation Of: Isoparametrische Hyperflächen In Sphären, Math. Ann. 251, 57–71 (1980) By Hans Friedrich Münzner., Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an English translation of the article "Isoparametrische Hyperflächen in Sphären" by Hans Friedrich Münzner, which was originally published in Math. Ann. 251, 57–71 (1980).

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper. I want to thank Cristina Ballantine for her help with the translation.

Translation Of: Isoparametrische Hyperflächen In Sphären Ii. Über Die Zerlegung Der Sphäre In Ballbündel, Math. Ann. 256, 215–232 (1981) By Hans Friedrich Münzner., Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an English translation of the article "Isoparametrische Hyperflächen in Sphären II. Über die Zerlegung der Sphäre in Ballbündel" by Hans Friedrich Münzner, which was originally published in Math. Ann. 256, 215–232 (1981).

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper. I want to thank Cristina Ballantine for her help with the translation.

Reviving The Media Station: A Study In Reverse Engineering, Nathanael Gentry, Melesa Poole

Liberty University Research Week

Undergraduate

Applied

Emerald Ash Borer And The Application Of Biological Control In Virginia, Shuheng Chen, Yihui Wu, Shengjie Liu

Arts & Sciences Student Symposium

The emerald ash borer (Agrilus planipennis; EAB) is an invasive wood-boring beetle whose larvae feed on ash phloem. After only 1-5 years of infestation, the larvae create extensive tunnels under the bark that disrupt the tree’s ability to transport water and nutrients, which eventually girdles and kills the tree. Since 2008, EAB has spread to all but the eastern-most counties in Virginia. Bological control is one strategy to limit EAB populations. In this project we study control by native agents (woodpeckers) and imported agents (parasitoid wasps).

Mathematical models of host-parasitoid interactions and simulations based on both models and field studies …

Reducing The Maximum Degree Of A Graph: Comparisons Of Bounds, Peter Borg

Theory & Applications of Graphs

Let λ(G) be the smallest number of vertices that can be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. Let λ_e(G) be the smallest number of edges that can be removed from G for the same purpose. Let k be the maximum degree of G, let t be the number of vertices of degree k, let M(G) be the set of vertices of degree k, let n be the number of vertices in the closed neighbourhood of M(G), and let m be the …

Road Network Detection Based On Improved Flicm-Mrf Method Using High Resolution Sar Images, Lamei Zhang, Chenwei Jiang, Bin Zou, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

The automatic detection of road network from satellite and aerial images is highly significant in many actual applications, for instance, urban traffic measurement, military emergency response, and vehicle target tracking. Compared with other high-resolution satellite remote sensing images, high-resolution synthetic aperture radar (SAR) has become a popular research perspective for road detection owing to its insensitivity to the atmosphere and sun-illumination. However, the method of road network detection is still lagging due to the strong multiplicative speckle noise and complex background interference, causing the loss and break in the road segment extraction results. Aiming to solve this problem, a three-step …

A High-Precision Sar Echo Simulation Method Based On Fdtd, Bin Zou, Yanqiu Jia, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

Synthetic aperture radar (SAR) echo simulation offers a low-cost and convenient way to obtain high-resolution images of targets, and plays an important role in system design and algorithm validation. Although high frequency approximation simulation is widely used, it is considered to be imprecise when calculating scattering field of fine structures, such as exhaust pipes and groove structures, especially in low frequency band. In this paper, a finite-difference time-domain (FDTD) based method is proposed for high-precision SAR echo simulation. In this method, scattering process of electromagnetic wave is accurately simulated to obtain equivalent electric and magnetic current on the surface of …

On The Third Moment Of L(½​, ΧD​) {Ii}: The Number Field Case, A. Diaconu, Ian Whitehead

Mathematics & Statistics Faculty Works

We establish a smoothed asymptotic formula for the third moment of quadratic Dirichlet L-functions at the central value. In addition to the main term, which is known, we prove the existence of a secondary term of size x^¾. The error term in the asymptotic formula is on the order of O(x^⅔+δ) for every δ > 0.

Teacher Educators Learning With Prospective Teachers: Finding Relevant Mathematics In Our (Their) Lives, Lindsay M. Keazer, Eryn Stehr Maher

Mathematical Sciences: Faculty Publications

Two mathematics teacher educators (MTEs) discuss the mathematical contexts generated by prospective teachers (PTs) when pushed to look for relevant mathematics in their lives and communities. Through collaborative teacher action research focused on iterations of collecting, categorizing, and discussing PTs’ mathematical contexts, and posing selected examples for PTs’ own examination, layers of learning occurred for both PTs and MTEs. PTs began to craft more personalized, story-like contexts, seemingly noticing more mathematics in their lives. MTEs were unexpectedly pushed to clarify their thinking about what it means to develop contexts that are authentic and relevant, and to contemplate how their actions …

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 …