Abstract: Many real-world phenomena tend to be modeled via nonlinear models. In the late 1980's, Ronald Mickens of Clark Atlanta University introduced the concept of a nonstandard finite difference scheme (NSFD) as a methodology which would best approximate solutions to systems of nonlinear differential equations. This talk will uncover how to construct NSFD schemes for various nonlinear models, including the well-known models for the spread of a disease models and the model for the dynamics of the infamous Tacoma Narrows Bridge.

Abstract: Numerical hydrodynamic forecasts are used in the atmosphere—for weather forecasts—and in bodies of water to predict biogeochemical variables as well as transport. Achieving the best forecasts requires both sophisticated mathematical modeling and methods for combining the modeled output with observational data. This talk will discuss how forecasts of chaotic and forced dynamical systems—such as atmospheres and oceans—are created, how these predictions are used for transport studies, such as quantifying the amount and distribution of plastic pollution in the Great Lakes, and how these ideas extend to problems beyond the Earth systems including cardiac electrophysiology.

In a popular paper of Cohn, the concept of a covering number of a group was introduced. The covering number of a finite group \(G\) is the smallest number of proper subgroups of \(G\) whose set-theoretic union is \(G\). Covering numbers are the subject of prior research by numerous authors, and in this talk we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. In addition, we will compare our results to some of the well-known results on covering numbers.

The permanent is a function on square matrices that computes a sum of products of entries in the matrix. This lesser-known function is more computationally complex than the determinant of a matrix. It has applications in combinatorics, quantum computing, and graph-theory. The Oppenheim Inequality states that the determinant of a Hadamard product of two matrices is bounded below by the product of their individual determinants. Bapat & Sunder conjectured that an analogue of this inequality would apply for permanents of matrices; however, 31 years later, the Bapat & Sunder conjecture was proven false upon discovery of a 7x7 counterexample of complex entries as well as a 16x16 counterexample of real entries. We investigate whether or not these counterexamples are minimal with respect to matrix size using techniques used to prove the conjecture true for 2x2 and 3x3 matrices. We discuss our results for 4x4 matrices with real and complex entries.

Assume that \(\mathbb{Q}\) is the field of rational numbers, \(\mathbb{R}\) is the field of real numbers, \(n\geq 2\) is a positive integer, and \(p\) is a prime number. Let \(a=\sqrt[n]{p}\in\mathbb{R}\) and \(F_{n}=\mathbb{Q}[a]\) be the extension field of \(\mathbb{Q}\) by \(a\). Then \(F_{n}=\{\alpha_{0}+\ldots+\alpha_{n-1}a^{n-1}\ | \ \alpha_{i}\in\mathbb{Q}\}\). Each element in \(F_{n}\) is a polynomial of \(a\) over \(\mathbb{Q}\) with degree less than \(n\). Define \(P{n}=\{\beta_{0} +...+ \beta_{n-1}a^{n-1} \ | \ \beta_{i}\in\mathbb{Q}^{+}\}\), where \(\mathbb{Q}^{+}=\{q\in\mathbb{Q} \ | \ q\geq 0 \mbox{ under the usual total order on}\mathbb{Q}\}\).
Conjecture: \(\forall f\in F_{n}\), there exists \(0 \neq g\in P_{n}\) such that \(fg\in P_{n}\).
We give a proof of the following special case.
Theorem: \(\forall f\in F_{5}\), there exists \(0\neq g\in P_{5}\) such that \(fg\in P_{5}\).

We build a program for portfolio analysis that uses the fundamentals of linear algebra and finance to calculate the predicted risk and return of a portfolio under different economic outlooks. This program can be used to identify assets that do not contribute positively to the portfolio and should therefore be removed. This model can be applied to a portfolio of any size using real historical data, such as for FAANG stocks (Facebook, Apple, Amazon, Netflix, and Google) and more.

Almost anything can be modelled with math, even fast food seasoning. For years crystallographers have investigated the atomic architecture of materials with a periodic structure such as quasi-crystals and sodium chloride (a.k.a. french fry seasoning). This talk will introduce the field of crystallography and explore the mathematics used to describe regular materials.

Our ancestors’ dream was to reach for the stars and rockets get us closer every day. As we reach further into space, our desire to explore the stars leads to the uncovering of how the universe came to be. In this presentation, we will discuss the history of rockets from Archytas in the 3rd century B.C. up through the current time period. We will also discuss the mathematics behind some of the aspects of rocket science.

We present here our search for the number of “legal” configurations and the explore the problem’s relation to the self-avoiding walks on the cubic lattice and string restrictions involving self-similarity and symmetry. Patterns, algorithms, and visualizations have been found in twisting corners of our path as we continue to shed light on the Rubik’s Snake.

Using surprisingly simple and accessible tools such as asking the right questions, irrational property, as well as the power of algebra one can gain some insight on how the \(\sqrt{2}\) can be utilized. With the curious behavior of \(\sqrt{2}\), we will convey how mathematical thoughts evolve from computation to general mathematical ideas.

Salute! is a game that helps children develop automaticity with their addition facts, it allows students to practice addition in an interactive way that is not spiritless. In this session, the game is adapted to the overlearning of multiplication facts. “Salute! To Multiplication” is a cognitive activity that helps third graders “recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts” according to the Texas Essential Knowledge and Skills. “Salute!” not only helps students to practice and master their multiplication skills but also division. The key to gaining automaticity is overlearning and while drilling multiplication facts can be boring playing a game is not. Playing this game has the students working on and memorizing their multiplication facts and appeals to a child’s natural urge to play and encourages them to learn more through playing this game and other cognitive activities. In addition to the presentation, the participants will have a chance to play the game during the session.

In language, codeswitching occurs when a speaker uses two or more languages in the context of one conversation. Speculation on motivation for switching is multifaceted; it is possible that a person may switch languages to hide certain information from listening native speakers, to better express themselves because certain words cease to exist in a given language, or to accommodate the person with whom they are speaking. Though it has been proposed that conversation dynamics affect factors of codeswitching, few works exist on the mathematical quantification of the hypothesized correlations between two individuals’ switching frequency.
In this work, we used machine learning, and specifically natural language processing, techniques to gain insights into code switching mechanisms by analyzing expert annotated codeswitching data. Specifically, we trained Naive Bayes classifier to predicting if codeswitching will occur based on various conversation features. The Naives Bayes classifier was chosen because it is a good baseline algorithm that is known to work well on similar problems, like spam prediction. Although predicting codeswitches is an inherently hard problem, we found that the features associated with an utterance of speaker A do have a predictive value for whether speaker B will codeswitch.

Clostridium difficile infection (CDI) poses a serious health threat for hospitalized patients. For an example in 2015 there were approximately 500,000 infections and 29,000 deaths in the United States. Individual risk factors include advanced age, prolonged antibiotic use, and severe illness. Environmental risk factors of CDI include bathroom sharing with infected patients, patient transfers, and seasonal trends. Despite several clinical trials testing various strategies for reducing CDI rates, most have been unsuccessful or provided short-term success. So, what we aim to do is effectively choose hospitals within a region to include in a clinical study with the goal of maximizing the reduction of CDI cases for a fixed treatment efficacy rate. To achieve this we used a linear mixed effects model to predict CDI rates 24 months out. Using this prediction model, we test various selection strategies such as targeting hospitals with high CDI rates, larger proportions of patients over 65, and high levels of centrality within the hospital network. Then using the selection methods we simulated interventions to determine which selection strategy resulted in the maximum reduction of CDI cases over 24 months. This work was done in correlation with the University of Iowa Department of Biostatistics through a program called ISIB, which was funded by National Heart, Lung, and Blood Institute.

The way a cognitive processing system handles multiple information sources can be classified in two ways. The first is the system’s architecture. There are two main types of architecture: serial, where information is processed sequentially, and parallel, where information is processed simultaneously. The second classification type is stopping rule, which asks whether the system stops after sufficient information is processed, or whether the system processed all sources of information until completion. The first of these is called self-terminating whereas the second is called exhaustive. Systems Factorial Technology (SFT) is a mathematical technique for classifying architecture and stopping rule from response time data in human cognitive tasks. We apply SFT to analyze how people process numerical information from the numerator and denominator when doing tasks with fractions. Using Mathematica, we compute two processing signatures for 7 different observers: the survivor interaction contrast and capacity curve. These curves both point to a serial self-terminating system, implying two things. First, people process fraction components one at a time, and second, people process only the necessary parts of the fraction that are needed for the task.

For ozone pollution studies, predictive pollutant concentration models have become indispensable tools to provide early warnings for an efficient air quality management and control system. Among these pollutants, the surface level ozone concentrations can be predicted using the Classification and Regression Trees (CART) method, which uses decision trees to find the best predictors for determining the value of a variable of interest. As has been done in a similar manner in another study, we have begun to apply the CART method to determine what atmospheric variables serve as the best predictors for identifying low, medium, and high ozone days in the Houston area using data collected from the Texas Commission on Environmental Quality.

Non-transitive dice can be described as a game with n dice that have unusual face values; the first n-1 players choose any dice, and the set of dice is non-transitive if the last player can always choose one of the remaining dice that has a greater than 50% chance of beating each of the other dice. This game is well-known with two players and small n but becomes more complicated as more players are introduced. In 1992, Oskar van Deventer found a set of seven dice for a three-player game. Although no such set of dice had been found, it is a known result that a four-player game would require 19 dice. Using specific properties of non-transitive dice such as quadratic residues and tournament graphs, and adapting a construction by Angel, Davis (2016), we were able to construct a set of 19 dice that can be played by four players. We will describe our construction and discuss ways to extend our research to other open questions about non-transitive dice.

Racial gerrymandering is drawing political maps to dilute the voting power of groups based on race. Texas Congressional District maps were recently contested at the Supreme Court on the basis of racial gerrymandering. Specifically, evidence suggests that Texas Congressional District 35 was packed with a higher proportion of Hispanic/Latinos than its surrounding area to dilute their voting power from neighboring districts like 21, which is less Hispanic/Latino than its surrounding area. To measure this, we use significance tests to compare the proportion of Hispanic/Latinos in each district to that of their convex hulls. We find that the probability of drawing districts with Hispanic/Latino proportions such as these from their convex hulls by chance is zero, suggesting that these districts were drawn with intent. We expanded this significance test to the rest of Texas Congressional Districts to compare each district to its convex hull. We do this to identify if districts 35 and 21 are unique circumstances or if this is normal occurrences throughout Texas.

Gerrymandering has often been used to suppress the votes of minority groups and limit the representation of these groups on a municipal level. However, gerrymandering can also be used to give minority voters a stronger voice in an election.
In this presentation, we will discuss the issue of gerrymandering and how mathematics can be used to approach this problem on a local scale. Specifically, we focused on city council districts in Seguin, TX. To study this, we produced a large number of potential district plans from an initial district template in order to observe the number of district plans that provide an opportunity for minority communities to achieve more equal representation.

Two of the most iconic results of voting theory are Condorcet's paradox and Arrow's impossibility theorem. We place these results in the context of Ramsey theory - the study of collections of sets that are partition regular. We give a new short proof of Arrow's theorem which uses Condorcet's paradox. Along the way the notion of an ultrafilter plays a prominent role. We present ultrafilters through the lens of Ramsey theory, i.e. in terms of partition regularity​.

Courts at all levels are struggling with the increasingly pressing and complex issue of political gerrymandering. Deadlines for the post-2020 census redistricting are quickly approaching. At the heart of our difficulties to fairly divide ourselves in voting district lies a math problem – how do we measure fairness? How can we use that measure to draw fair district boundaries?
Our project is part of nationwide collaboration of mathematicians, demographers, lawyers, mapmakers, political leaders, and citizens attempting to develop tools for this purpose. We will survey Markov Chain Monte Carlo methods used in the PA Supreme Court case and our recent work to improve and apply it to more states. We will discuss several commonly used compactness metrics and present a new idea called transit time compactness that aims to use the Google Maps API to measure cohesiveness of people, not just land.
This talk derives from participation in the Voting Right Data Institute at MIT, Harvard, and Tuft Univ in Summer 2018 under Dr. Moon Duchin.

Borda Count: A voting system with a flaw During this session we will teach the Borda Count method. The borda count method is a voting system and just like any other voting system it has a critical flaw. We will be explaining how it violates the IIA (independence of irrelevant alternative) and explore how it can be manipulated to receive the outcome we desire. We will demonstrate choices of foods then give each person a ballot and let the audience choose their preferred order. We will then put people in pairs and allow them to manipulate their ballot against their opponent to receive their preferred outcome.

The Trojan Y-Chromosome (TYC) strategy is a proposed eradication method for an invasive species. The strategy includes a one time introduction of a modified male, or supermale with two Y chromosomes so that the male to female ratio is skewed. Numerical approximation is used to explore the efficacy of introducing supermales periodically coupled with suppression of the natural population at the time of introduction.

A model of SIR (Susceptible, Infected and Recovered) will be developed to examine the Epstein-Barr virus and the disease infectious mononucleosis on a U.S. college population. Mononucleosis is a common disease in developed countries and 35% to 50% of college students will get mono. We will present a numerical overview and sensitivity analysis.

Graphing three-dimensional equations can be done in several different ways. The method chosen was through Marching Cubes. This talk will discuss how the Marching Cubes algorithm works to construct three-dimensional surfaces, using a grid of points, given the domain of each axes.

A conic section is an intersection of a plane and a cone creating different shapes. General conic sections include circles, ellipses, parabolas, and hyperbolas. This presentation investigates the persistent homology of a specific class of conics equations, namely a circle.We show how to find the zeroth and first homology of any circle in standard form, \((x-h)^2 + (y-k)^2 = r^2\). These ideas lead to further research including other conic sections such as ellipses, parabolas, and hyperbolas.

We will explore the Fourier expansions for functions of two variables. We will first introduce the concept of the traditional Fourier Series expansion for some classic wave forms such as the square wave and sawtooth function. Then using the complex form of the Fourier expansion for these wave forms, we will show how to extend to three dimensions. That is, we derive the Fourier coefficients for some real-valued functions of two variables and illustrate with the resulting three-dimensional surfaces.

I'm going to show how to create a music composition based on the Klein Group.

We will derive a general equation for the first homology of a Lissajous pillow; a parametric curve defined by \(f(t)=(cos(at) , sin(bt))\). We will start by explaining what homologies are and then transition into what they mean to us on the two-dimensional plane by giving a few quick examples on simple functions. This will then lead into how we can apply these same ideas to parametric functions such as the Lissajous pillow.

Famous philosopher Pythagoras, who is well known with his theorems in math, had also contributed to music theory about the frequency of vibration. In the seventeenth century, Mersenne proved Pythagoras’s theorem on pitch and string tension wrong through his research. After so many centuries, Mersenne showed that the true relation between tension and frequency of vibration was his formula for the fundamental frequency and not what Pythagoras thought. In this speech, I will show Mersenne’s formula and discuss its implications in music theory and applications in strings used in musical instruments with a simulation program.

Pascal’s Triangle and Pascal’s Pyramid both have symmetric properties that relate to probability. In this talk, we will explore their connection by taking “random” steps as part of the chaos game. We will then explore the resulting structure or fractal, referred to as Sierpinski’s Triangle and its 3D extension, Sierpinski’s Pyramid.

Active-learning techniques have become very popular in the past decade or so. The East Texas Inquiry-Based Learning in Mathematics Consortium is recruiting people interested in active-learning techniques, whether an experienced practitioner or someone interested in honing their skills, to provide a support structure for teachers and instructors throughout our region. We will discuss possible first events, and gather information and ideas from interested people during this session.

SET is an addictive game with many hidden mathematical properties still being discovered. The SET Universe is the entire 81 cards in a deck of the game SET, where most rules of geometry remain but some are altered due to the structure of the game. Points, lines, and planes are well defined in the SET Universe and can be counted. A variation of the game introduces the concept of “planets” and their properties within the SET Universe. We count the number of planets in the SET Universe and the probability of having a planet in four randomly drawn cards.

We examine artificial intelligences playing Mrs. Pac-Man under partially observable conditions. The agents are generated algorithmically and are represented by neural networks. These topics are described from first principles. We will measure the similarities and differences of these networks to see the number of fundamentally different solutions.

Take a string, tangle it up, and fuse the ends. You have just made a knot, a mathematical object of which there are an infinite number. Distinguishing between different knots is a complex mathematical problem, and a diverse array of mathematical disciplines prove useful in finding different ways to represent and encode information about a knot that remains unchanged no matter how a knot is presented in space. Some such properties, termed invariants, can include, but are not limited to, polynomials, matrices, topological groups, and objects in hyperbolic geometry. Our research focuses on finding new polynomial knot invariants which we will present in our talk. Additionally, we will describe the methods we used to calculate these invariants and explain their relation to other well-known knot polynomials.

We study the effects of certain local moves on HOMFLYPT and Kauffman polynomials. We show that all HOMFLYPT (or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

We develop a formula for defining the geodesic on the surface of a cube. While there are algorithms for geodesics on the surface of convex polyhedra, our approach differs in that we define a formula that will yield the shortest path simply given the coordinates of any two arbitrary points on the cube. In addition, we develop a formula for finding any point along the shortest path. It is worth noting that the process outlined in this talk could be extended to the remaining regular polyhedra, and possibly even to any convex polyhedron.

Stereographic projection is an interesting technique for mapping the surface of spheres to planes. It is angle-preserving, which gives rise to noteworthy characteristics, such as circles maintaining their shapes through the mapping. Due to this, it is occasionally used in cartography to map the surface of planets so that craters remain circular. A visual 3D model will be integrated into the presentation to make the concepts more understandable.

We want a model that will optimally distribute limited scientific funding among its constituent fields in a way that produces the most science. Our dynamical systems approach is based on author productivity derived from their papers on the ArXiV. An author is funded by a field in proportion to their contributions to that field. The author then uses this funding to produce more science (not limited to that field), which will update the funding distribution for the next cycle. We will find the stable, optimal distribution based a rough estimate of current real-world funding levels.​

To provide sufficient satellite coverage for Antarctica research bases, a satellite constellation must be designed that provides continuous coverage with minimal cost. Flower constellations and other highly-elliptical orbits have been proposed for coverage over high-latitude regions, yet complications involving radiation and quality of connection have made it difficult to identify an optimal constellation. In this talk, we will discuss the mathematics behind highly-elliptical orbits and flower constellations, use geometry to compare the coverage of each, and propose a new constellation using polar orbits.

This proposal investigates the improvement of estimation in the field of survey sampling, where estimation is equivalently known as the attachment of weights to the survey statistics. A well-known technique of this estimation is to adjust the design weights and to calibrate those weights based on optimization of a metric measure subject to certain constraints on the auxiliary information available at the estimation stage. The proposed approach will emphasize on the weight system and derive a distance procedure with the support of calibration equations. This research will review existing calibration techniques and will propose new improved calibration techniques for different situations. This study examines the notion of new calibrated estimators that will be compared with its competitors analytically as well as through simulation studies.

In the shadow of the Rubik’s Cube lay the Rubik’s Snake, a twistable chain of triangular prisms. The Snake presents us a combinatoric conundrum: Each segment can turn four ways, but the Snake fills space as it goes and cannot pass through itself. While commercial versions of the snakes tout “endless possibilities”, can this be true? Of course not!

Chikungunya is a viral infection that is spread by mosquitoes of the genus Aedes. Chikungunya victims experience symptoms similar to those caused by the Dengue and Zika viruses but are less likely to die from it. Chikungunya was not a major research interest up until 2004 when a third of the population on Reunion Island, located in Africa, was infected by the disease. This outbreak inspired the creation of mathematical models to study how Chikungunya is transmitted. Driven by these studies, we constructed a game-theoretic model that considers how rational individuals decide to use mosquito repellent to prevent the disease. Typical epidemiological models are constructed using a system of ordinary differential equations. Our model comprises of this kind of system, but is also coupled with a game-theoretic model. Individuals make their decision based on a payoff function that is affected by the steady-state values of the system. We found that at Nash equilibrium individuals use mosquito repellent more if the perceived cost of using mosquito repellent is decreased in respect to the consequence of contracting the disease. However, setting the perceived cost of mosquito repellent to zero does not guarantee the eradication of Chikungunya. With this in mind, governments and disease control programs can better understand how to manage the transmission of this mosquito-borne disease.

RNA is a single-stranded molecule that can fold onto itself according to rules given by the Watson-Crick base pairs. We explore a mathematical abstraction of this process where the pairing relations can be given by any graph and foldings are modeled by non-crossing matchings on a word. Using this, we define when two bonding graphs are equivalent and classify graphs acting on sequences of length less than 8.

Singular Value Decomposition is a matrix operation that has so many uses and applications, specially in data storage and approximation. In this project, which started from and focuses on computational results, we experimented with an SVD-based process called the best rank-one approximation, and applied it to the incidence matrix of path graphs. In this paper we narrate how we found these interesting features displayed in these approximation matrices, and later elaborate on our attempt at understanding them from a more theoretical point of view.

The neighborhood designs strongly regular graphs such as triangular, lattice and Payley graphs are used to create linear codes, of which the latter have been investigated heavily within the past decade. Within this project, we add to previous results by exploring circulant graphs, in order to discover binary linear codes \(C_n(S)\) created from circulant line graphs. In doing so, we describe special classes of codes separated by the properties of the circulant graphs that produced them; hence, providing the relationship of the graphs to their influence of the parameters of the resulting binary linear codes.

Lyme disease is one of the most prevalent and fastest growing vector-borne bacterial illnesses in the United States, with over 25,000 new confirmed cases every year. Lyme disease cases have more than doubled in recent years and the Centers for Disease Control and Protection estimates that those numbers could be significantly underrepresented. Humans contract the bacteria, Borrelia burgdorferi, through the bite of Ixodes scapularis, commonly known as the Deer Tick or Eastern Blacklegged Tick. The tick can receive the bacterium from a variety of small mammal and bird species but white-footed mice are the primary reservoirs in the northeastern United States, especially near human settlement. The life cycle and behavior of the ticks depends greatly on the season, with different stages of tick biting at different times. Reducing the infection in mice populations and the overall tick population can greatly reduce the number of humans affected by this disease. We use a mathematical model of transmission dynamics and vector demographics to see to what extent two different tick control methods, oral vaccination of white-footed mice and application of acaricides to the mice's fur through specially designed bait boxes, can affect the population of infected I. scapularis. Using a system of difference equations to model both tick and mice populations, we determine effectiveness of these methods implemented either together or separately.

The Trojan Y Chromosome eradication strategy is a means of ridding a habitat of an invasive species. Sex treatment on a wild member of the population skews their typical chromosomes so that their progeny is strictly male. In attempts to throw off the sex ratio of the population, the species is predicted to be driven towards extinction. Here, the traditional model of the TYC using sex treated "supermales" undergoes linear analysis and numerical approximations. An adjusted model goes through the same process. Evaluations of the results suggest further modification of the model to accurately assess the efficacy of the strategy.

Reusable diapers are growing in popularity due to new and improved designs. The total cost associated with reusable diapers is the sum of the upfront cost of purchasing the diapers and the reoccurring cost of cleaning the diapers. With an application of the AM-GM inequality we show that the minimum cost to the consumer is twice the geometric mean of the upfront and cleaning costs. We conclude with an analysis of our reusable diaper model and a cost comparison with disposable diapers.

The Trojan Y Chromosome (TYC) eradication strategy is a proposed method that attempts to manipulate the overall ratio of males to females for a given organism’s population to drive down the total amount. All processes in nature are fundamentally stochastic, so by adding in noise to the TYC model we will be able to produce more realistic results. With this amount of variability, we can now provide confidence intervals for different initial conditions to ensure a relatively high chance of eradication.

We will develop an evolutionary algorithm for the structural changes of galvanized steel beams over time under external forces. These beams will be used to construct lattice tower models. These models will treat the galvanized steel beams as ideal springs with limits in both tensile and compressive forces. We will apply an evolutionary algorithm to optimize the tensile and compressive strengths of the beams. By modeling structures in this way, we can create a stable collection of beams that are optimized to withstand specified external forces, without the cost of over-engineering. Because of the large number of calculations and the need for multiple trials to determine success, we plan to use parallel processing on high end graphics cards to execute these simulations. This method will allow us to rapidly design more efficient structures.