Student Presentations
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Undergraduate Students Talks
- Elizabeth Andreas: Two-Dimensional Neuron Modeling
- Caroline Bradley: Research On Hill Cipher
- John Brotemarkle: A Time Delay Model of Immune Response with Drug Resistance in Tumor Cells
- Vince Campo: Dog Vaccinations and Quarantine: A Mathematical Approach on Rabies
- Kyle Cerrito: A Sangaku is Worth a Thousand Words: Rediscovering Japanese Puzzle Pictures
- Chris Chamberlin and Jacob DeCapua: Towards a Game Analogue of Brooks' Theorem
- Katherine Craven: n Disks Walk onto a Peg
- Elijah DeJonge: Destroying the Ramsey Property by the Removal of Edges
- Aidan Draper: Investigating Image Quality Loss
- Sara Feggeler: Bifurcation Diagrams for a Logistic Growth Model with Grazing
- Joshua Fitzgerald: To Boldly Go: A Python Framework for Automatic Low-Energy Trajectory Design
- Spencer Gales: The Mathematics of the Head Injury Criterion
- James Grubbs: A Study of Towers of Hanoi
- Jackson Leonard: Bifurcation Diagrams for Population Dynamics Models with Weak Allee effects
- Mary McBride: Dynamics of an HIV-1 Virotherapy Model
- Luis Ramirez: Applying Spectral Projected Gradient and Simulated Annealing Optimization for CT Image Reconstruction
- Michael Reed: Optimal Polynomial Form Characteristic Methods
- Jessica Stevens: A Mathematical Model for Tumor Growth and Treatment Using Virotherapy
- Will Stiles: Anti-Ramsey Hypergraph Numbers
Undergraduate Students Posters
- Kenyona Bethea and Shantaga Lemasney: Optimal Personal Protection Strategies for Middle East Respiratory Syndrome
- Alex Chandler: Categorifications from Thin Posets
- Cameron Cinel: Extensions of the $p$-adic numbers of degree $p^2$
- Hannah Elser and Dana Gerraputa: On Game Chromatic Number: Mycielskians
- Addie Harrison: Visual Human Tracking and Image Formation
- Carolina Herrera and Ciera Tucker: Optimal Condom Usage to Prevent Chlamydia
- Hannah Ross and Kristina Smith: Hand Foot Mouth Disease - Fuzzy Game Theoretic Approach
- Lynn Huang: An Algorithm for Adjusted Kernel Linear Discriminant Analysis
- Juan Quiroa: Enumeration of Ramification Polygons of Degree $p^2$
- Jaime Seith: Tuning Parameter Selection for KLDA
- Philip Smith: Transfer Learning with Deep CNNs for the Purposes of Gender Recognition and Age Estimation
- Cyera Taylor: Joint Estimation of Age, Gender, and Race using different Feature Extraction Methods on Morph-II
- Hanwen Wang: Expected Time of Escape and Most Probable Paths for Transitions between Metastable States
- Jacob Zoromski: Degree $mp$ Extensions of the $p$-Adic Numbers
Graduate Students
- Alex Chandler: Thin Posets and Diamond Transitivity
- Nalin Fonseka: Analysis of Positive Solutions for a Logistic Growth Model with Grazing
- Ivanti Galloway: Measuring the Cost of Avoiding Vaccinations
- Elliott Hollifield: Numerical Bifurcation Curves for a non-local Boundary Value Problem
- Austin Lawson: Persistence Curves: A New Vectorization of Persistence Diagrams
- Qing Liu: Anisotropic functional deconvolution with long-memory noise
- Aaron Rapp: Dual-Wind Discontinuous Galerkin Methods with Applications to Obstacle Problems
- Maximilian Rezek: Why Are Leaves Wavy?
- Susan Rogowski: Modeling DNA Replication using the sine-Gordon Equation
- Joshua Safley: Stability of Dishonest Signalling in Extended Pygmalion Game
- Victor Summers: An application of spectral sequences to the Khovanov homology of 3-braids
- Kalani Thalagoda: Continued Fractions with Irrational Denominators
- Claire Zajaczkowski: Surgery Obstructions to Seifert Fibered Homology Spheres
Two-Dimensional Neuron Modeling
Elizabeth Andreas, University of Washington - Tacomamentored by Dr. Duong (Rita) Than
Abstract: The Morris-Lecar model originally used to describe the electrical pulses in the muscle fiber of a barnacle is quantitatively similar to the behavior of a squid axon used by Hodgkin and Huxley to model the propagation of neuron signals. Due to this, the two-dimensional Morris-Lecar model can be used as a reduced version of the four-dimensional Hodgkin-Huxley model. The reduced, single cell, model is a powerful tool used to understand the topological behaviors of action potentials, such as the shape and the onset of limit cycles. In this presentation, the use of phase portraits and neural dynamics will be used to analyze action potentials. As well as why neurons switch between two modes of operation, single signal and oscillatory propagation.
Optimal Personal Protection Strategies for Middle East Respiratory Syndrome
Kenyona Bethea and Shantaga Lemasney, Bennett Collegementored by Dr. Igor Erovenko and Dr. Ajanta Roy
Abstract: Middle East Respiratory Syndrome (MERS) is a global threat and has the potential to cause large outbreaks with substantial public health and economic costs. It is caused by a single-stranded RNA virus, called a coronavirus. The virus is transmitted from human-to-human and one of the most effective strategies to prevent the disease is usage of protective gear (i.e. masks and gloves) in nosocomial settings. With no vaccination or cure, MERS has become a growing concern. In this project, we present a game-theoretical model in which individuals choose to either use protective gear or not.
Research On Hill Cipher
Caroline Bradley , Gardner-Webb Universitymentored by Dr. Olga Poliakova
Abstract: The main purpose of this paper is to research Hill cipher including its modifications over the years. The paper also explores modern uses of the cipher. Hill cipher in its original form is not widely used because of its vulnerability to many different types of attacks. However, with modifications it is used in many technological advances in recent years. Learning about Hill cipher and its history and origin can yield the understanding of more complex uses of the cipher.
A Time Delay Model of Immune Response with Drug Resistance in Tumor Cells
John Brotemarkle and Genia Kennedy, Winthrop Universitymentored by Dr. Kristen Abernathy and Dr. Zachary Abernathy
Abstract: Drug resistance, also known as multidrug resistance (MDR), is the leading cause of chemotherapy failure in treating cancer. This drug resistance in cancer cells can be transferred from resistant cancer cells to sensitive cancer cells. Sensitive cancer cells can become resistant through three main methods: direct cell to cell contact with resistant cancer cells, through a membrane, or through exposure to the treatment drug. In our project, we take into account the transfer of drug resistance from resistant to sensitive cancer cells via direct cell to cell contact. We then introduce an immune response and chemotherapy, and establish conditions on treatment parameters in the resulting system to ensure a globally stable cure state. We provide evidence of a limit cycle and conjecture the existence of a Hopf bifurcation. Furthermore, we consider the effects of a delay on the immune response and numerically demonstrate how such a delay can cause further bifurcations of internal equilibria.
Dog Vaccinations and Quarantine: A Mathematical Approach on Rabies
Vince Campo and John Palacios , University of Guammentored by Dr. Hyunju Oh and Dr. Hideo Nagahashi
Abstract: China, the world's second leading country in Rabies, is an area of interest when analyzing the lethal disease. The paper moves to create a strategy to effectively combat the disease by means of dog vaccinations and quarantine. Using data based in China, the paper models the dynamics of the disease and proposes a model to help better understand the how the disease is spread. Within is an analysis of the threshold points of disease behavior to understand what conditions affect the transmission of Rabies. Throughout, is a mathematical overview on different scenarios of Rabies, when it is introduced and also when it has become an endemic. Lastly, by utilizing Game Theory the paper offers a strategy for individual dog owners on separate methods of battling Rabies, vaccine and quarantine where only one method is present. The paper plays on the idea of having a dominant strategy to battle Rabies when certain costs are met.
A Sangaku is Worth a Thousand Words: Rediscovering Japanese Puzzle Pictures
Kyle Cerrito, Lenoir-Rhyne Universitymentored by Dr. Sarah A. Nelson
Abstract: There are many powerful theorems and puzzles whose proofs can be shown by pictures. In this talk, I will discuss some examples of Sangaku puzzles which Japanese mathematicians created to decorate temples and shrines in the 1700s and 1800s. These pictures challenge the reader to prove unique geometric theorems and puzzles using geometric concepts like similar triangles, complementary and supplementary angles, and, of course, the Pythagorean Theorem; however, while the tools to solve such puzzles may be simple, the answers are far from obvious. I will share my results from successfully and unsuccessfully proving Sangaku theorems without looking at existing proofs. Also, I will demonstrate some geometric properties I discovered in the process.
Towards a Game Analogue of Brooks' Theorem
Chris Chamberlin, Jacob DeCapua, Hannah Elser, Dana Gerraputa, Winthrop Universitymentored by Dr. Arran Hamm
Abstract: A graph is a collection of vertices and edges. A proper coloring of a graph is an assignment of color to each vertex so that no edge has the same color on both ends. One major result in the study of graph coloring is Brooks' Theorem which bounds the number of colors required to properly color a graph by the maximum number of edges at any of its vertices. Shifting gears, consider the following game played by Alice and Bob on a graph. The players will take turns (with Alice going first) coloring the vertices from a common color set so that no edge has its two vertices colored the same color (i.e. after each player's turn, the partial coloring is proper). Alice wins if the entire graph is colored and Bob wins otherwise. Thus the question: given a graph G and a color set C, who has a winning strategy (i.e. can win independently of the other player's moves)? This problem was introduced in the 1980's and is a direct generalization of graph coloring. In this talk, we will discuss progress toward obtaining an analogue in the game coloring setting of Brooks' Theorem.
Thin Posets and Diamond Transitivity
Alex Chandler, NC State Universitymentored by Dr. Radmila Sazdanovic
Abstract: A partially ordered set (poset) is a set endowed with a reflexive, antisymmetric, transitive relation. Intuitively, a poset is a set for which there is a notion of one element being `less than' another, but not all elements need be comparable in this way. An example to keep in mind is the collection of subsets of some finite set, where the partial order is given by set containment. Finite posets are visualized with a directed graph called the Hasse diagram, with vertices and directed edges corresponding to elements of the set, and minimal `less than' relations. A poset is called thin if sufficiently short intervals are forced to be diamond shaped. In this talk, we consider an action of diamonds in a thin poset on directed paths in the Hasse diagram. We give sufficient conditions which guarantee any two directed paths between common endpoints in a Hasse diagram of a thin poset can be taken to one another by actions of diamonds. Such posets are called diamond transitive. Diamond transitivity is useful for those who study functors on posets since it significantly reduces the conditions one must check to guarantee such a functor is well defined.
Extensions of the $p$-adic Numbers of Degree $p^2$
Cameron Cinel, University of Southern Californiamentored by Dr. Chad Awtrey, Dr. Sebastian Pauli, and Dr. Scott Zinzer
co-authored by Judah Devadoss, Elisabeth Howard, Alex Jenny
Abstract: Let $p$ be a prime. We determine the isomorphism classes of totally ramified extensions of the $p$-adic numbers of degree $p^2$. For each extension, we determine a defining Eisenstein polynomial, automorphism group, and ramification polygon. This generalizes results previously given by Amano (1971) and Awtrey-Hadgis (2017), who gave similar classifications for the case of $p$ and $2p$, respectively.
n Disks Walk onto a Peg
Katherine Craven, Lenoir-Rhyne Universitymentored by Dr. Timothy Goldberg
Abstract: The Towers of Hanoi is a classic puzzle that involves moving disks from one peg to another, according to certain rules. The player can only move one disk at a time and can never place a bigger disk on top of a smaller one. In this talk, I will discuss the original puzzle and then describe my work on a related puzzle, where the pegs are arranged in a triangle and disks can only be moved in a fixed direction between adjacent pegs. I will relate my result on an upper bound for the number of moves required to solve this puzzle, and outline ideas for narrowing this upper bound.
Comparing Prediction Errors Associated with Data Splitting Methods
Jacob Crouse Elon Universitymentored by Dr. Laura Taylor
Abstract: Researchers developing statistical models oftentimes chose a data splitting method, which separates data sets into training and testing sets, without thoroughly considering alternative methods. This research compares the predictive accuracy of simple linear regression models created using simulated data sets containing different amounts of standard error. Additional data sets with the assumption of normality violated are introduced to further assess the reliability of data splitting methods and prediction errors. Data splitting methods used in this research include dividing 50% of the data into a training set and 50% into a testing set, 10-fold cross validation, and leave-one-out cross validation. Several measures of prediction error are calculated to assess model performance including Mean Square Prediction Error, Mean Square Simulated Error, Mean Absolute Error, and Root Mean Square Error. Preliminary findings suggest Root Mean Square Error and Mean Absolute Error tend to underestimate prediction error when an assumption violation is introduced to the data set. This emphasizes the need for researchers to consider which prediction error they use to assess model performance on a case-by-case basis.
Destroying the Ramsey Property by the Removal of Edges
Elijah DeJonge, Western Carolina Universitymentored by Dr. Mark Budden
Abstract: The Ramsey Number of two graphs $G$ and $H$, denoted $R(G,H)$, is the least natural number $n$ such that every red-blue coloring of the edges of a complete graph on $n$ vertices must contain a red subgraph isomorphic to $G$ or a blue subgraph isomorphic to $H$. Inherent in this definition is the idea that there exists a red-blue coloring of a complete graph on $n-1$ vertices that does not contain a red subgraph isomorphic to $G$ or a blue subgraph isomorphic to $H$. In this sense, the removal of a vertex and all of its $n-1$ incident edges from a complete graph on $n$ vertices destroys the Ramsey Property. However, it may not be necessary to remove all $n-1$ edges incident with a vertex to construct a red-blue coloring that lacks a red $G$ or a blue $H$. In this talk, we will investigate how many edges must be removed to destroy the Ramsey Property for various pairs of graphs.
Investigating Image Quality Loss
Aidan Draper , Elon Universitymentored by Dr. Laura Taylor
Abstract: Statisticians, as well as machine learning and computer vision experts, have been studying image enhancement through denoising different domains of photography, such as textual documentation, tomography, astronomical, and low-light photography. With the surge of interest in machine- and deep-learning, many in the computer vision field feel that current approaches for effective image denoising are moving away from statistical inference methods and, instead, moving into these subfields of artificial intelligence. However, this paper sheds some light on current applications that show how statistical inference-based methods and frameworks that rely on conditional probability and Bayes' theorem are still prevalent today. We reconstruct a few inferential kernel filters in the R and python languages and compare their effectiveness in denoising RAW images. In doing so, we also surveyed Elon students about their opinion of a single filtered photo using the various methods. Many scientists believe that noise filters cause blurring and image quality loss so we investigated whether or not people felt as though improved denoising causes any quality loss as compared to their original images. Individuals assigned scores indicating the image quality of a denoised photo compared to its true no-noise image on a 1 to 10 scale. Survey scores for the various methods are compared to benchmark tests, such as peak signal-to-noise ratio, mean squared error, and r-squared, to determine if there were any correlations between visual scores and the benchmark results.
On Game Chromatic Number: Mycielskians
Hannah Elser and Dana Gerraputa, Winthrop Universitymentored by Dr. Arran Hamm
Abstract: A graph is a collection of vertices and edges. A proper coloring of a graph is an assignment of color to each vertex so that no edge has the same color on both ends. The chromatic number of a graph is the least amount of colors that must be used in order for the graph to have a proper coloring. The Mycielski construction on a graph results in a new graph which requires one additional color to properly color. Shifting gears, consider the following game played by Alice and Bob on a graph. The players will take turns (with Alice going first) coloring the vertices from a common color set so that no edge has its two vertices colored the same color (i.e. after each player's turn, the partial coloring is proper). Alice wins if the entire graph is colored and Bob wins otherwise. Thus the question: given a graph G and a color set C, who has a winning strategy (i.e. can win independently of the other player's moves)? Our work focused on finding a relationship between the game chromatic number of a graph and that of its Mycielskian.
Bifurcation Diagrams for a Logistic Growth Model with Grazing
Sara Feggeler, UNC Greensboromentored by Dr. Ratnasingham Shivaji
coauthored by Nalin Fonseka, Dr. Byungjae Son
Abstract: We study positive solutions to the two point boundary value problem which models describes the steady states of a population that is governed by logistic growth and encounters grazing by a predator. We discuss the bifurcation diagrams of the positive solutions to this model. We also study the evolution of the bifurcation diagram as the exterior matrix hostility varies.
To Boldly Go: A Python Framework for Automatic Low-Energy Trajectory Design
Joshua Fitzgerald, Greensboro Collegementored by Dr. Stuart Davidson
Abstract: Traditional space trajectory design methods use two-body models, which only simulate one spacecraft and one planet or star at a time. They consequently generate fast but fuel-inefficient and imprecise pathways through space. Space trajectory design methods that use three-body models, which simulate one spacecraft and two masses at a time, can produce ""low-energy"" trajectories. Compared to traditionally derived trajectories, low-energy trajectories are slower but are more accurate and are much more fuel-efficient. The goal of this project is to produce a streamlined Python program, tentatively named To Boldly Go, capable of generating low-energy trajectories automatically. To Boldly Go will automatically use the tube manifold approach to low-energy trajectory design to find initial position-velocity conditions that, when integrated, will correspond to an itinerary provided by the user. It will operate within the context of a Planar Circular Restricted Three-Body Problem, a straightforward special case of the general three-body problem, with user-defined parameters. Development has followed an iterative design methodology and an approximately functional programming paradigm.
Analysis of Positive Solutions for a Logistic Growth Model with Grazing
Nalin Fonseka, UNC Greensboromentored by Dr. Ratnasingham Shivaji
coauthored by Dr. B. Son (Wayne Sate University) and K. Spetzer (UNCG)
Abstract:We study positive solutions to a steady state reaction diffusion equation that models a population that is governed by logistic growth and grazed by a predator. We discuss nonexistence, existence, multiplicity and uniqueness results. Our existence and multiplicity results are obtained via the method of sub-super solutions.
The Mathematics of the Head Injury Criterion
Spencer Gales, Wake Forest Universitymentored by Dr. John Gemmer
Abstract: The Head Injury Criterion (HIC) score is used as a predictor for the outcome of a head injury during front impact collision and is widely used in North American motor vehicle safety regulations, the assessment of playground safety, and in the assessment of athletic helmets. In particular, the HIC score measures local rapid fluctuations in the acceleration curve during an accident to predict whether a serious injury will occur. However, the HIC score as it is currently formulated is ill-posed in the sense that acceleration curves that minimize the HIC score is a discontinuous function. This result is physically unsatisfactory as it leads to infinite jerk (divergent third derivative). Our research consists of studying a regularized version of this functional that penalizes sharp deviations in the jerk. The goal of the research is to reformulate the HIC score in a manner that smooths out the divergent third derivative and to mathematically prove that it is possible to minimize this modified version of the HIC score.
Measuring the Cost of Avoiding Vaccinations
Ivanti Galloway, UNC Greensboromentored by Dr. Stephen Robinson
Abstract: On April 10, 2017, The Minnesota Department of Health received notice about a suspected measles case in an unvaccinated child. On April 11th, there was a second report of the measles. On April 13th, a third case was confirmed. By May 31, 2017, 65 cases of the measles, an infectious disease thought to be eliminated in the United States, were confirmed. This outbreak occurred in an area with a low vaccination rate. Improvements in medical care have far removed many in the United States from risks of such infectious diseases. Consequently, this removal of illness has given way to the anti-vaccination movements as well as the idea of spacing out vaccinations. In this research we investigate a system of ordinary differential equations that model the effect of anti-vaccination groups on the spread of infectious diseases, such as the measles, in a society as well as quantify the cost that avoiding vaccination and disease impose on a society.
A Study of Towers of Hanoi
James Grubbs, USC Salkehatchiementored by Dr. Wei-Kai Lai
Abstract: In our independent study class, we studied the Towers of Hanoi. I will share in this talk some findings we had in this class and demonstrate some solutions with minimum number of moves in each case. We start with the traditional Towers of Hanoi along with its mathematics. We then explore several of its variations, including the version with more than 3 poles, the version with different moving rules, and the version with poles in different heights. Some of its use in popular culture will also be introduced.
Visual Human Tracking and Image Formation
Addie Harrison, Wake Forest Universitymentored by Dr. John Gemmer
Abstract: Animals in motion have the ability to track an object with their vestibular ocular system in order to keep the image stationary on the retina. A motivating example is a chicken's ability to keep their head stationary despite the forces that are being applied to their body. In our work, we are interested in understanding the mathematics of image formation and image tracking. Using tools from vector calculus, projective geometry and linear algebra, we developed a model that maps images from space to the retina. This model allowed us to create optical flow vector fields that shows the path light takes on the retina. Furthermore, using differential equations, we developed a dynamic 3D model of the head coupled to the eyes which track an object in space. In collaboration with neuroscientists we can use our models to learn how the visual system and the brain are used in conjunction to stabilize an image as well as to process depth perception.
Optimal Condom Usage to Prevent Chlamydia
Carolina Herrera and Ciera Tucker, UNC Greensboromentored by Dr. Igor Erovenko and Dr. Ajanta Roy
Abstract: Chlamydia is an infectious disease spread by the bacteria Chlamydia Trachomatis that affects millions of people worldwide. The disease is spread through sexual contact between an infected and susceptible individual. Protective measures include abstinence and long-term monogamy, but the most common and simple method is condom usage. In this project, we took a game theoretic approach to figuring out the optimal protection level needed to eradicate Chlamydia with condom usage as our protective measure of choice. We performed a sensitivity analysis to demonstrate the effects of changing parameter values on the optimal solution of the system. We found in our study that the total eradication of the disease by individual selection of protection is not possible.
Numerical Bifurcation Curves for a non-local Boundary Value Problem
Elliott Hollifield, UNC Greensboromentored by Dr. Maya Chhetri
co-authored by Dr. Petr Girg, University of West Bohemia
Abstract: We present numerical results for a non-local non-linear boundary value problem. We use a finite element method together with a nonlinear solver to find positive weak solutions of a fractional Laplacian boundary value problem in 1 dimension. We construct numerical bifurcations curves and demonstrate how the Laplacian operator can be recovered by the fractional Laplacian.
An Algorithm for Adjusted Kernel Linear Discriminant Analysis
Lynn Huang, Iowa State Universitymentored by Dr. Cuixian Chen and Dr. Yishi Wang
Abstract: The current implementation of KLDA in R fails to compute projections in the case that the kernel matrix is non-invertible in the objective function. In this presentation, we propose an algorithm for adjusted KLDA which allows for the approximation of singular matrices within KLDA's objective function, ensuring the success of computations for any set of tuning parameters. The validity of the algorithm is evaluated on several simulated datasets, then applied to three versions of a subset of the Morph-II dataset containing different extracted features for face imaging tasks. The transformed feature set is used to train several statistical classification models, whose performance is then evaluated to determine the efficacy of the algorithm.
Persistence Curves: A New Vectorization of Persistence Diagrams
Austin Lawson , UNC Greensboromentored by Dr. Yu-Min Chung
Abstract: Topological Data Analysis (TDA) is a field of mathematics concerned with analyzing the shape of data to extract information. The main tool of TDA is Persistent Homology, which is used to generate a summary of the data called a Persistence Diagram. Recently, research in the field has been focused on combining TDA and machine learning through a vectorization of these diagrams. we propose a class of such vectorizations called Persistence Curves. By combining these curves with the support vector machines algorithm, we achieve high accuracy scores on two texture datasets, Outex and UIUCTex.
Bifurcation Diagrams for Population Dynamics Models with Weak Allee effects
Jackson Leonardmentored by Dr. Ratnasingham Shivaji and Dr. Jerome Goddard (Auburn University-Montgomery)
co-authored by Nalin Fonseka
Abstract: We will discuss existence, uniqueness, and multiplicity results for a steady state reaction diffusion equation that models a population governed by an Allee effect growth rate and U-shaped density dependent dispersal on the boundary of the habitat. We obtain the bifurcation curve for positive solutions via a quadrature method and Mathematica computations.
Anisotropic functional deconvolution with long-memory noise
Qing Liu , Ohio Universitymentored by Dr. Rida Benhaddou
Abstract: We look into minimax results for the anisotropic two-dimensional functional deconvolution model with multi-parameter fractional Gaussian noise. We derive the lower and upper bounds under the Lp-risk, and construct a wavelet hard-thresholding estimator that attains asymptotically quasi-optimal rates in a wide range of Besov balls. The convergence rates depend on a balance between the parameters of the Besov balls, the degree of ill-posedness of the convolution and the parameters of the long-range dependence in all directions. Taking advantage of the Riesz poly-potential, a wavelet-vaguelette expansion is applied to de-correlate the anisotropic fractional Gaussian noise. A limited simulation study in the 2-dimensional case confirms theoretical claims of the paper.
Dynamics of an HIV-1 Virotherapy Model
Mary McBride, Connor Hennessy, G. Tarque, Winthrop Universitymentored by Dr. K Abernathy and Dr. Z Abernathy
Abstract: In this project, we consider the dynamics of the HIV-1 virus under the effects of virotherapy and an immune response. We calculate basic reproductive ratios for the HIV-1 virus and recombinant virus, and use these ratios to establish existence and stability criteria for disease-free, single-infection, and double-infection equilibria. We utilize Lyapunov functions to prove the global asymptotic stability of the disease-free and single-infection equilibria. For the double-infection equilibrium, we explore its stability through numerical simulations and provide evidence of a Hopf bifurcation. We conclude with a discussion on the effects of using a recombinant virus to control HIV-1-infected cell populations.
Enumeration of Ramification Polygons of Degree $p^2$
Juan Quiroa, UNC Greensboromentored by Dr. Sebastian Pauli and Dr. Chad Awtrey
co-authored by Alex Jenny (Midwestern State University)
Abstract: From Krasner's Lemma, we know that there are only finitely many non-isomorphic extensions of the field of $p$-adic numbers of a given degree; hence a complete classification is possible. The type that we chose for this project was totally ramified extensions of degree $p^2$. Each extension is generated by a Eisenstein polynomial, which has a unique $j$-value. In addition, each set of such polynomials that share a $j$-value also share the same ramification polygon, which is an invariant of such extensions. Our goal was to enumerate and characterize all distinct ramification polygons for a given $j$-value and throughout all extensions of the aforementioned type. Furthermore, upon graphing such relation we observe that there is a parabolic pattern in the size of set of distinct ramification polygons and $j$-values.
Applying Spectral Projected Gradient and Simulated Annealing Optimization for CT Image Reconstruction
Luis Ramirez, William M. O'Keeffe and Bradford H. Halter, University of Washington Bothellmentored by Dr. Thomas Humphries and Dr. Milagros Loreto
Abstract: We apply the hybrid optimization method of Guerrero (et al.) to CT image reconstruction. CT image reconstruction can be posed as the problem of minimizing $|Az - b| + \lambda T(z)$, where $T(z)$ is a Tikhonov regularization term to account for ill-conditioning of the Hessian. The hybrid method uses the spectral projected gradient method to compute the optimal regularized solution, along with simulated annealing to simultaneously determine the regularization parameter lambda. The infinity norm of the spectral projected direction is used by simulated annealing as the energy function to determine a suitable value of lambda. Through numerical experimentation, we find that the energy function used by Guerrero (et. al) does not give the desired results for CT image reconstruction, and that 2-norm regularization does not improve image quality. We therefore investigate using total variation (TV) and the Huber prior as regularization terms, as well as alternative energy functions. While both regularizers improve image quality, numerical issues make it difficult to adapt the hybrid method to use TV. We therefore use the Huber prior, and find a new energy function which is used to determine lambda. Our modified hybrid approach is shown to significantly improve image quality in several numerical experiments.
Dual-Wind Discontinuous Galerkin Methods with Applications to Obstacle Problems
Aaron Rapp, UNC Greensboromentored by Dr. Tom Lewis and Dr. Yi Zhang
Abstract: A discontinuous Galerkin (DG) finite-element interior calculus has been developed as a common framework to describe various DG approximation methods for second-order elliptic problems. In this presentation, we will discuss the dual-wind DG method and its application to the obstacle problem with Dirichlet boundary conditions. In particular, we will consider applying the method to the problem $-\Delta u \ge f$ on $\Omega$ with $u=g$ on $\partial \Omega$ and $u\ge \psi$ on $\Omega$, where $\psi$ is the given obstacle. We will also present the results for several numerical tests that validate the performance of the method.
Optimal Polynomial Form Characteristic Methods
Michael Reed, UNC Greensboromentored by Dr. Thomas Lewis
Abstract: There are several possible forms for a polynomial expression terms, such as factored form, expanded form, Horner form, and many others. The set of such possible forms is an equivalence class of a universal algebra. However, when the operations are replaced with floating point pseudo-operations, the equivalence relation on the terms no longer holds and a rounding error is introduced. For each polynomial the computational effort and numerical error can be minimized with a form that is equivalent on the dense polynomial algebra by using a comparison of characteristic values. A local and a global theory of characteristic values is investigated to select the optimal form on a floating point interval.
Why Are Leaves Wavy?
Maximilian Rezek, Wake Forest Universitymentored by Dr. John Gemmer
Abstract: In the non-Euclidean model of elasticity, growth is modeled by a Riemannian metric that encodes local changes in distance. In response to the growth, the sheet deforms to minimize an elastic energy. The elastic energy consists of the sum of the stretching and bending energy. Minimizers of the stretching energy consist of isometric immersions of the metric, while minimizers of the bending energy remain flat. The competition between bending and stretching selects a pattern in the sheet. In this talk, we will show that periodic patterns have the lowest energy for a large class of metrics. Qualitatively, our results agree with patterns observed in leaves and torn elastic sheets.
Modeling DNA Replication using the sine-Gordon Equation
Susan Rogowski, Wake Forest Universitymentored by Dr. Sarah Raynor
Abstract: DNA replication begins when local unwound regions of several broken hydrogen bonds form. These regions are often referred to as bubbles and their formation can be modeled using a chain of coupled pendulums. The motions of angular oscillations that occur are usually modeled using the sine-Gordon equation. In this model, the discrete analog of the sine-Gordon equation is derived. Instead of taking the continuous limit, the space step between pendulums remains constant and the forces on the pendulum by Newton's laws of motion are considered. Additionally, the randomization of nitrogen bases, Adenine, Thymine, Guanine, and Cytosine, within the chain of pendulums is modeled by randomly selecting the corresponding coefficients for each base. The system of differential equations is solved and plotted using Verlet integration in MATLAB.
Hand Foot Mouth Disease - Fuzzy Game Theoretic Approach
Hannah Ross and Kristina Smith, Bennett Collegementored by Dr. Igor Erovenko and Dr. Ajanta Roy
Abstract: In this project, we present a fuzzy game theoretic approach to Hand-Foot-Mouth disease. Hand-Foot-Mouth-Disease (HFMD) is an acute contagious disease, one that has affected many populations around the world. It is caused by viruses belonging to the enterovirus genus (group). The group of viruses includes polio viruses, coxsackie viruses, echo viruses, and entero viruses. Although there are several viruses, coxsackie virus A16 and enteroviruses 71 (EV71) are the most common to produce HFMD. Precise estimate of disease transmission rate is critical for epidemiological Game theory model. We considered the transmission coefficient to be uncertain and described by a (symmetric) triangular fuzzy number. We defuzzified the optimum results by a signed distance method. We studied sensitivity analysis to demonstrate the effects of changing parameter values on the optimal solution of the system. We compared the crisp and fuzzy model. We concluded that if the uncertainties are accounted for in an appropriate manner, that would lower the basic reproduction number. Also, we discussed the effects on Nash equilibrium.
Stability of Dishonest Signalling in Extended Pygmalion Game
Joshua Safley , UNC Greensboromentored by Dr. Jan Rychtar
Abstract: We present a signalling game based on the Pygmalion game, played between a population of Senders and a population of Receivers. Senders of varying qualities $q$ send signals of strength $s$ and die with probability $1 - t(s,q)$. Furthermore, this signal can be destroyed by the environment with probability $r(s)$. Receivers can detect signal strengths but cannot detect the quality of an individual sender. We show that honest signalling is often the best strategy for Senders but that in Nash equilibria only dishonest signalling occurs. Interestingly, however, in the signalling equilibria Receivers face a higher proportion of high-quality individuals than in non-signalling pooling equilibrium.
Tuning Parameter Selection for KLDA
Jaime Seith, Lynn Huang, Jackson Maris, Ryan Wood, NC State Universitymentored by Dr. Yishi Wang and Dr. Cuixian Chen
Abstract: Kernel Linear Discriminant Analysis (KLDA) is a dimension reduction technique that can be used on large datasets to project the data into a lower-dimensional feature space and achieve an optimal separation of classes. The difficulty using this method comes with selecting an optimal tuning parameter that will separate the data without over-fitting the model. In this paper, a loss function is defined by examining the classification error rates in Support Vector Machines and the radius of the data cloud to determine an optimal tuning parameter. The loss function is then minimized after applying it to a two-class toy dataset. More investigation is needed to apply it larger datasets and to multiple-class KLDA problems. Further research is also needed to define the loss function without discontinuities in order to find the exact minimum globally. With these adjustments, this loss function can be used universally on other dimension reduction techniques to improve classification accuracy and selection of models that best-fit the dataset in focus.
Transfer Learning with Deep CNNs for the Purposes of Gender Recognition and Age Estimation
Philip Smith, UNC Wilmingtonmentored by Dr. Cuixian Chen
Abstract: In this project, a competition-winning deep neural network with pretrained weights is used for image-based gender recognition and age estimation. Transfer learning is explored by testing the effects of changes in various design schemes and training parameters in order to improve prediction accuracy. Training techniques such as input standardization, data augmentation, and label distribution age encoding are compared. Finally, a system is tested that first classifies subjects by gender, and then uses separate male and female age models to predict age. This paper shows that, with proper training techniques, good results can be obtained by retasking existing convolutional filters towards a new purpose.
A Mathematical Model for Tumor Growth and Treatment Using Virotherapy
Jessica Stevens , Winthrop Universitymentored by Dr. Zach Abernathy and Dr. Kristen Abernathy
Abstract: We present a system of four nonlinear differential equations to model the use of virotherapy as a treatment for cancer. This model describes interactions among infected tumor cells, uninfected tumor cells, effector T-cells, and virions. Using various stability analysis techniques, we establish a necessary and sufficient treatment condition to ensure a globally stable cure state. We additionally show the existence of a cancer persistence state when this condition is violated and provide numerical evidence of a Hopf bifurcation under estimated parameter values from the literature. We conclude with a discussion on the biological implications of our results.
Anti-Ramsey Hypergraph Numbers
Will Stiles, Western Carolina Universitymentored by Dr. Mark Budden
Abstract: An $r$-uniform hypergraph $H$ consists of a non-empty vertex set $V$ and a set of hyperedges $E$ whose elements are unordered $r$-tuples of elements from $V$. A rainbow $H$ is a coloring of the hyperedges of $H$ such that every hyperedge receives a distinct color. For $n \ge |V|$ the anti-Ramsey number $ar_n(H)$ is the maximum number of colors that can be used in any hyperedge coloring of the complete $r$-uniform hypergraph $K_n^{(r)}$ without producing a rainbow copy of $H$. In this talk we will consider anti-Ramsey numbers for certain hypergraph paths and complete hypergraphs missing a single hyperedge.
An application of spectral sequences to the Khovanov homology of 3-braids
Victor Summers NC State Universitymentored by Dr. Radmila Sazdanovic
Abstract: Spectral sequences are powerful homological algebraic tools which have seen widespread application to low-dimensional topology. Khovanov homology is an invariant of knots and links that takes the form of a bigraded abelian group. In this talk, I will give an application of spectral sequences to the structure of torsion - elements of finite order - in the Khovanov homology of a class of links known as 3-braids.
Joint Estimation of Age, Gender, and Race using different Feature Extraction Methods on Morph-II
Cyera Taylor, Salem Collegementored by Dr. Cuixian Chen
Abstract: In 2014, Guo and Mu introduced a framework to jointly estimate age, gender, and race by using regression techniques including linear and non-linear canonical correlation analysis and partial least squares regression. They considered the feature extraction method of bio-inspired features (BIFs) on Morph-II. This study extends Guos experiment with the use of different feature extraction methods such as histogram oriented gradients (HOGs) and local binary patterns (LBPs), in addition to BIFs to see which extraction methods are the most effective at jointly estimating age, gender, and race.
Continued Fractions with Irrational Denominators
Kalani Thalagoda, UNC Greensboromentored by Dr. John Greene (University of Minnesota Duluth)
Abstract: A continued fraction is a cascading set of fractions obtained by an iterative process. A continued fraction is called a simple continued fraction when the numerator is equal to 1 and non-simple otherwise. Simple continued fractions of square roots of natural numbers has particularly nice patterns. In this study, our goal was to understand non-simple continued fractions of a generalized square roots.
Expected Time of Escape and Most Probable Paths for Transitions between Metastable States
Hanwen Wang, Wake Forest Universitymentored by Dr. John A Gemmer
Abstract: In this poster, we use the large deviation principle of Freidlin-Wentzell to understand noise induced tipping events in stochastic differential equations. Specifically, we focus on a model of eutrophication in a lake and a consumer resource model. The Freidlin-Wentzell theory allows us to compute the most probable path to escape the basin of attraction of a stable point. The time of escape can also be estimated by Laplace's method. Our results are then compared with Monte-Carlo simulations generated using an Euler-Maruyama discretization.
Surgery Obstructions to Seifert Fibered Homology Spheres
Claire Zajaczkowski, NC State Universitymentored by Dr. Tye Lidman
Abstract: We examine surgery on a knot in the three sphere to determine surgery obstructions to Seifert fibered integral homology spheres. Dehn surgery is one of our key ways of understanding 3-manifolds, and Seifert fibered integral homology spheres are a class of manifolds we understand well. Thus it is a well explored topic to find such surgery obstructions. In this talk we will find such surgery obstructions using Heegaard Floer and Knot Floer homology, which has been a commonly used approach in the past. Here however, we take a different approach and use the number of singular fibers of a Seifert fibered integral homology sphere to find obstructions, which is the toroidal structure. This approach led us to some significant and new results by looking at the genus of the knot and the number of singular fibers.
Degree $mp$ Extensions of the $p$-Adic Numbers
Jacob Zoromski, University of Wisconsin - Madisonmentored by Dr. Chad Awtrey, Dr. Sebastian Pauli, and Dr. Scott Zinzer
co-authored by Kaitlyn McGloin
Abstract: Let $p$ be prime. We classify all totally ramified extensions of the $p$-adic numbers of degree $mp$, where $m < p$, $gcd(m,p) = q^n$ or $1$. We determine the discriminant valuation, a defining Eisenstein polynomial, and the automorphism group for each extension. Our research can be seen as a continuation of the work of Amano (1971) and Awtrey-Hadgis (2017), who determined classifications for extensions of degree $p$ and $2p$ respectively.