The 10th Annual UNCG Regional Mathematics and Statistics Conference
Saturday, November 1, 2014

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    1. Date: 11/01/2014
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Abstracts of the talks

This page will start to be updated in August and then about once a week, so do not worry if your name will not appear here right after you register.

    Undergraduate Students

  1. Matthew Buhr: Trade-off of Glial Cell Actions in Response to Ischemic Hypoxia in the Brain
  2. Kira Crawford and Aaleah Lancaster: Mathematical Model of African Sleeping Sickness
  3. Chasity Dorsett and Marie Paulemond: Dynamic Model of Dengue Fever
  4. Madeline Edwards: Identification of Transcription Factor Programs under Methylation Control during Brain Development
  5. Lee Fisher: A Geometric Approach to Voting Theory
  6. Robin French: A new algorithm for Galois groups of quintic polynomials
  7. Nicole Soltz: Counting Roots and Galois Groups
  8. David Suarez: Influence of Individual's Mobility and Neighborhood Size on the Evolution of Cooperation


    Graduate Students

  10. Aida Briceno: The Dynamics of Offensive Messages in the World of Social Media
  11. Ontario Stotts: Isomorphy Classes of Trivolutions of $SL_2(k)$
  12. Jia Zhao: 3D Mathematical Modeling and Simulations of Cell Mitosis by a Phase Field Approach


The Dynamics of Offensive Messages in the World of Social Media

Aida Briceno, University of North Carolina at Greensboro, Greensboro, NC
mentored by Dr. Sherry Towers and Dr. Kamuela Yong

Abstract: The 21st century has redefined the way we communicate, our concept of individual and group privacy, and the dynamics of acceptable behavioral norms. The messaging dynamics on Twitter, an internet social network, has opened new ways/modes of spreading information. As a result cyberbullying or in general, the spread of offensive messages, is a prevalent problem. The aim of this report is to identify and evaluate conditions that would dampen the role of cyberbullying dynamics on Twitter. We present a discrete-time non-linear compartmental model to explore how the introduction of a Quarantine class may help to hinder the spread of offensive messages. We based the parameters of this model on recent Twitter data related to a topic that communities would deem most offensive, and found that for Twitter a level of quarantine can always be achieved that will immediately suppress the spread of offensive messages, and that this level of quarantine is independent of the number of offenders spreading the message. We hope that the analysis of this dynamic model will shed some insights into the viability of new models of methods for reducing cyberbullying in public social networks.

Trade-off of Glial Cell Actions in Response to Ischemic Hypoxia in the Brain

Matthew Buhr, University of South Dakota, Vermillion, SD
mentored by Dr. Carlos Castillo-Chavez

Abstract: A model that explores the dual role of glial cells in the formation of scar tissue and in neural repair, following hypoxia ischemia in the brain, is introduced. We consider multiple pieces of evidence and assumptions that include the fact that scar tissue helps protect the brain during the acute phase of insult by limiting the spread of secondary damage caused by the ischemia; that scar tissue limits recovery by inhibiting both the formation of connections between cells and the repair of damaged neurons; and that repaired neurons offer better functional recovery, but do nothing to halt the spread of ischemic injury. A stochastic, spatially explicit cellular automaton model is used as our insilicate version of the brain's biology. Mathematical analysis is carried out via mean field (MFA) and pair approximation (OPA) models. We show that the MFA, which neglects all spatial autocorrelation, leads to brain death or a brain composed of only damaged and scarred neurons. The OPA demonstrates that explicitly modeling spatial adjacency gives outcomes that include the possibility of scar tissue surrounding damaged cells (preventing further secondary death). Our results inform how the trade-off between scar tissue formation and neural repair impacts future brain health. This supports the use of mixed-treatment regimes.

Mathematical Model of African Sleeping Sickness

Kira Crawford and Aaleah Lancaster, Bennett College, Greensboro, NC
mentored by Dr. Hyunju Oh Dr. Jan Rychtář

Abstract: We will present a mathematical SIR compartment model pertaining to African sleeping sickness using Tsetse vectors to a multi-host population. The proportion of cattle being treated with insecticide-treated cattle (ITC) per day is taken into effect to control Tsetse flies and Trypanosoma brucei rhodesiense. There is an analytical expression used for the basic reproduction number in the presence of ITC treatment using the hosts (cattle, humans, and pigs). The sensivity was observed based on the basic reproduction number in regards to the three hosts in the presence of ITC treatment. A result has shown that the basic reproduction number is more sensitive to changes in the Tsetse morality and the proportion of cattle treated with insecticides per day. It will induce a minimum cost for controlling T.b rhodesiense.

Dynamic Model of Dengue Fever

Chasity Dorsett and Marie Paulemond, Bennett College, Greensboro, NC
mentored by Dr. Hyunju Oh Dr. Jan Rychtář

Abstract: Dengue Fever is one of the most important vector-borne infections that have emerged as a worldwide problem beginning in the 1950s. This infection is transmitted by its principal vector Aedes Stegomyia aegypti. In this paper, we will present a SIR compartment model that includes Human as the host, Mosquitoes as the vector, and Eggs as the vertical Transmission. From these basic models, we are trying to eliminate the spread of Dengue Fever by killing all eggs and mosquitoes. From these SIR compartment models we deduced the basic reproduction number R0 and force of infected, λ for Dengue Fever. For elimination of the disease we find that the Basic Reproduction number, R0, is sensitive to the proportion of infected eggs, g, natural death rate for mosquitoes, M, and daily rate of biting from the mosquitoes, a. If infection of eggs is more than 83%, epidemic starts. The optimal solution is when the proportion of infected eggs, g, is at a maximum of 50% and minimum natural death rate for mosquitoes is at 0.3. The optimal result provides the minimize cost using Dichlorodiphenyltrichloroethane (DDT) and the reduce risk of using DDT for populations at risk.

Identification of Transcription Factor Programs under Methylation Control during Brain Development

Madeline Edwards, Elon University, Elon, NC
mentored by Dr. Hehuang Xie

Abstract: Transcription factors (TFs) often bind to specific DNA sequences to promote or block gene expression. The interactions between TFs and target DNA sequences may be regulated by DNA methylation. It has been well recognized that DNA methylation plays an important role in neural differentiation, which is determined by a cascade of TFs. However, the epigenetic regulated TF programs critical to brain development remain largely unexplored. To fill in such a knowledge gap, we first analyzed mammalian brain methylomes to identify genomic loci differentially methylated during development. We compiled a set of experimentally validated TF binding sites from TRANSFAC 7.0, JASPAR 2014, and UniPROBE databases, and applied HOMER to identify TFs of which binding sites enriched in genomic regions hypomethylated in neurons and glial cells, respectively. Using MEME Suite and ClusterZ, we then determined pairs of TFs with binding sites frequently overlapped. With the Cytoscape program, we created a network of possible TF interactions, which can either be complex formation or regulatory. This predicted network provides novel insight to the epigenetic regulation controlling brain development.

A Geometric Approach to Voting Theory

Lee Fisher, Appalachian State University, Boone, NC
mentored by Dr. Vicky Klima

Abstract: Arrow's theorem informs us that while some voting methods may be more fair than others, a certain set of reasonable fairness conditions cannot simultaneously be satisfied by any one voting system. Well-known voting systems include the plurality method, where voters are told to pick just one candidate and the candidate who receives the largest number of votes wins; and the Borda count, where candidates are ranked and given points in equally-spaced descending order. This presentation proposes a new voting system which functions as a generalization of the Borda count. In our new system, we assign a value of one to first place, zero to last place, and intermediate rankings are assigned values in arbitrary intervals of non-increasing order. We then use geometric methods to consider the winning results space associated with each candidate. We will compare this voting system against the fairness criteria stated in Arrow's theorem. We will show this voting procedure is monotone, equivalent to the Borda count only in the three-candidate case, and like both the plurality and Borda count methods does not necessarily satisfy the Condorcet winner criterion; which selects the candidate who wins each pairwise comparison to win the election.

A new algorithm for Galois groups of quintic polynomials

Robin French, Elon University, Elon, NC
mentored by Dr. Chad Awtrey

Abstract: Finding solutions of polynomial equations is a central problem in mathematics. Of particular importance is the ability to solve a polynomial ``by radicals''; i.e., using only the coefficients of the polynomial, the four basic arithmetic operations (addition, subtraction, multiplication, division), and roots (square roots, cube roots, etc.). For example, the existence of the quadratic formula shows that all quadratic polynomials are solvable by radicals. In addition, degree three polynomials and degree four polynomials are also solvable by radicals, which was shown in the 16th century. However, the same is not true for all degree five polynomials. Therefore, we are left with the following question: how do we determine which degree five polynomials are solvable by radicals? To answer this question, we study an important object that is associated to every polynomial. This object, named after 19th century mathematician Evariste Galois, is known as the polynomial's Galois group. The characteristics of the Galois group encode arithmetic information regarding its corresponding polynomial, including whether or not the polynomial is solvable by radicals. In this talk, we will discuss a new algorithm for determining the Galois group of a degree five polynomial.

Counting Roots and Galois Groups

Nicole Soltz, Elon University, Elon, NC
mentored by Dr. Chad Awtrey

Abstract: Let $f(x)$ be an irreducible polynomial over a field $F$ with roots $a$, $b$, and $c$ (in some algebraic closure), and let $K=F(a)$ be the extension obtained by adjoining one root of $f$ to $F$. We present two methods for determining the Galois group of $f(x)$. One involves answering the question: is $(a-b)(a-c)(b-c)\in F$? The other involves answering the question: how many roots of $f$ are in $K$? We end by discussing an application to computing Galois groups of cubic polynomials defined over an extension of the $p$-adic numbers.

Isomorphy Classes of Trivolutions of $SL_2(k)$

Ontario Stotts, Wake Forest University, Winston Salem, NC
mentored by Dr. Loek Helminck

Abstract: Trivolutions are group automorphisms of order 3. $SL(2,k)$ is the group of 2-by-2 matrices with determinant 1. In this paper we will give a characterization for the isomorphy classes of trivolutions of $SL(2,k)$ with $k$ any field of characteristic not 2 nor 3. This work is analogous to work by Helminck and Wu on automorphisms of order 2.

Influence of Individual's Mobility and Neighborhood Size on the Evolution of Cooperation

David Suarez, University of North Carolina at Greensboro, Greensboro, NC
coauthored by Praveen Suthaharan (NC State) and Elizabeth Bergen (Cornell University)
mentored by Dr. Jan Rychtář and Dr. Jonathan Rowell

Abstract: Researchers have long sought to identify which mechanisms can overcome basic selection pressures and support the establishment of cooperation within a population. In a slight variation of their traditionally conception, cooperators are individuals who pay a cost for another individual to receive a benefit, whereas wildtype individuals are free-riders or defectors who receive but do not provide benefits, thus paying no cost. In models of well- mixed populations, free-riders are favored; however, cooperation can emerge and evolve in spatially structured populations. In this paper we extend previous results on the evolution of cooperation to a system of a finite, fixed population of mobile competitors. We conduct a stochastic simulation study examining how parameters controlling an individual's mobility and neighborhood size influences the likelihood that cooperation can evolve within the population. We find that both greater mobility and larger neighborhood size inhibit the evolution of cooperation because it allows the free-riders to find the cooperators faster and exploit them more.

3D Mathematical Modeling and Simulations of Cell Mitosis by a Phase Field Approach

Jia Zhao, University of South Carolina, Columbia, SC
mentored by Dr. Qi Wang

Abstract: During a cell cycle, mitosis is a process, in which a mother cell duplicates into two generically similar daughter cells. In the initial stage of mitosis, the mother cell, attached on a substrate, would undergo a dramatical shape change by detaching from the substance and forming a round surface. At the late stage of mitosis, a contractile ring would form in cell orbit and the mother cell would split into two daughter cells, which is known as cytokinesis for eukaryotic cells. Recently, we have developed a series of three-dimensional hydrodynamic models by a phase field approach, studying cellular mitosis. Qualitatively patterns of cell rounding, blebbing and division process have been captured, which agrees with experiment observation from our biologist collaborations. In this talk, our preliminary study on the mechanism and controlling factors of cell mitosis would be present. 3D numerical simulations will be shown, as well.


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