Abstract: In this talk, Dr. Butler will give an overview of his company, Daniel H. Wagner, Associates, from its historic beginnings in 1963 through its current role today. In 1963, Dr. Dan Wagner founded his eponymous company with two guiding principles in mind. First, Dan believed in hiring young mathematicians and training them to solve real-world problems. Second, Dan felt the quality of the writing in the technical reports and briefings provided to the clients was nearly as important as the technical content itself. To this end, and while the company size was small, Dan personally reviewed every scrap of paper that went out under the company name. Through the years, the company developed an impressive reputation for mathematical analysis applied to the budding field of Search Theory (find the lost H-bomb, find the sunken treasure, find the enemy submarine, etc.), and this continues to be an area of expertise today. At the same time, the company demonstrated the breadth of its capabilities by working in areas as diverse as DNA sequencing, retirement planning, crane anti-sway, speech recognition, speaker verification, and random number generation on GPUs.

Abstract: Applications of immediate concern, such as graph drawing and computer chip layout problems, random graph theory and modeling the internet, graph connectivity measures and ecological systems, etc. have driven some of the most interesting questions in the field of graph theory. Currently, scientists are engineering self-assembling DNA molecules to serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis. Often, the self-assembled objects, e.g. lattices or polyhedral skeletons, may be modeled as graphs. Thus, these new technologies in self-assembly are now generating fascinating and challenging new design problems for which graph theory is a natural tool. We will present some new applications in DNA self-assembly and describe some of the graph-theoretical design strategy problems arising from them. We will see how finding optimal design strategies leads to developing new algorithms for graphs, addressing new computational complexity questions, and finding new graph invariants corresponding to the minimum number of components necessary to build a target structure under various laboratory settings. This is a rich area for open problems, interdisciplinary collaborations, and involvement for all levels of investigators, especially students.

In the max-linear system, we define the tropical semiring \((\mathbb{\bar R}, \oplus, \otimes)\) by \(\mathbb{\bar R} := \mathbb{R} \cup \{- \infty\}\), \(a \oplus b := \max(a,b)\), \(a \otimes b := a + b\). Similarly, given matrices \(A = (a_{ij})\), \(B = (b_{ij}) \in \mathbb{\bar R}^{n\times n}\), \(i,j \in \{1, 2, ..., n\}\), we have \((A \otimes B)_{ij} = \bigoplus_{k}(A_{ik} \otimes B_{kj}) = \text{max}_k(A_{ik} + B_{kj})\). Tropical linear algebra has been studied for a wide range of applications in various fields, such as scheduling problems, discrete event systems, control theory, statistical inference and pairwise ranking. However, the question of when two matrices commute remains a mystery. This presentation investigates the geometric properties of a special case of the two-sided system given by tropical commuting constraints, as well as its relation to graph theory. Given a finite matrix \(A \in \mathbb{R}^{2\times 2}\), we look at the space spanned by the vectors of entries of \(B\) such that \(A \otimes B = B \otimes A\) and explore the set of extreme vertices that generates this tropical polyhedral cone up to tropical scaling.

Calculating powers of a defective matrix can be considerably difficult for arbitrarily large matrices. It was recently shown that for any defective matrix there exists an l-lift which embeds the matrix inside a larger, diagonalizable matrix, whose powers are easily calculable. This particular lifting, however, uses Lidskii's theorem to create the diagonalization which consequently perturbs the eigenvalues. Furthermore, the lifting does not allow for powers of the original matrix to be extracted from powers of the lifted matrix. We show an alternate way of lifting a defective matrix using some of its generalized eigenvectors such that the lifted matrix retains the eigenvalues of the original and also allows for one to extract the powers of the original from the powers of the lifted matrix.

When computer graphics first needed to be modeled, we used Euler angles, however, we face limitations such as Gimbal lock when attempting to animate these models, resulting in rougher motion. Quaternions, first discovered by William Hamilton, provide a more elegant solution for rotations and will never cause Gimbal lock. The unit quaternion space is also almost equivalent to the set of rotations in three-dimensional space. We will discuss the disadvantages of Euler angles in computer animation, and demonstrate why quaternions are the superior choice.

We consider the filtering set of noncommutative rings: \[\left. H = \{ R[x,y,z] \middle/ (x^2 = y^2 = z^2 = xyz = -1)\},\right.\] where \(R\) is a commutative ring with unity. We know that the Lipschitz and Hurwitz integers are elements of \(H\). The Lipschitz and Hurwitz integers share familiar properties with the integers, hence their names. For the Hurwitz integers in particular, there exists a notion of a prime factorization since there is a notion of divisibility amongst its elements.

Let \(f(x) = x^{3} - q_{2}x^{2} - q_{1}x - q_{0} \in \mathbb{Q}[x]\) be an irreducible polynomial with \(q_{0} > 0\). Then there exists a positive real number \(a \not \in \mathbb{Q}\) such that \(f(a) = 0\). We consider the total order on cubic extension field \(R = \mathbb{Q}[a]\) induced from \(\mathbb{R}\).

Let \(\mathbb{R}^{+} = \{r \in \mathbb{R} \ | \ r \geq 0\}\) and \(\mathbb{Q}^{+} = \mathbb{Q} \cap \mathbb{R}^{+}\). We want to find a way to characterize \(R^{+}\). Define \[ P = \{\alpha_{0} + \alpha_{1} a + \alpha_{2} a^{2} \ | \ \alpha_{i} \in \mathbb{Q}^{+}\}, \] and \[ F = \Big\{\frac{f}{g} \ | \ f, g \in P, g \neq 0\Big\} \subseteq R. \]

We prove that for certain conditions, \(R^{+}=F\). Without these conditions, \(F\subseteq R^{+}\) but \(F\neq R^{+}\). Examples are provided.

Statistical studies predict that up to 50% of the stars in our galaxy are part of a binary star system: a pair of stars orbiting around a common center of mass. Due to their abundance and unique characteristics, such systems are invaluable sources of astrophysical data. We are concerned with contact binaries, namely W-UMa systems: a pair of main sequence stars in physical contact sharing a common envelope. Due to mass transfer between the stars, the structure and evolution of these systems differ greatly from single-star systems. In this study, our goal is to develop an N-body model of a W-UMa system. With this, we explore the role of binary mass transfer in the formation of W-UMa subtypes and generate statistics that can be compared with well-studied systems.

In the US alone, transmission towers number in the tens of millions; most are lattice-based, comprising a framework of beams connected at nodes. At tens- to hundreds of thousands of dollars each, primarily in material costs, they are a sizeable expense to taxpayers and transmission companies.

The dilemma central to their optimization is that a beam's cross-sectional area is proportional to both its strength and cost, i.e. a thicker beam, while able to support a given load with less strain, will cost more and weigh more heavily on the beams supporting it.

By varying the cross-section of each beam, we want to make towers as light and inexpensive as possible without sacrificing their structural integrity. Known as truss-sizing optimization, this problem is differential in nature and heavily dependent on tower geometry, lending it to a computational approach.

Drawing inspiration from the atrophy and hypertrophy of muscles, we develop and evaluate an optimization algorithm that adaptively resizes beams based on their stress – a process that produces rapid results and allows the application of both static and dynamic loads, setting it apart from popular algorithms in this intensely studied field.

We will be presenting the basic knowledge and importance of the heat-kernel coefficients for a Laplace operator in three dimensions with Dirichlet boundary conditions. We will show how tools from complex analysis, the gamma function, and the spectral zeta function may be used in the computation of these coefficients. The purpose of our research is to evaluate the computation time and effectiveness of this method in comparison to others.

The Trojan Y Chromosome Method is a strategy to eradicate invasive species. To carry this method out, a Trojan-male fish, called a supermale, is released into the environment. Mating between wild-type females and the supermales is guaranteed to be male. After one generation, the male to female sex ratio will be skewed, causing the female population to decline and, ideally, leading to extinction to invasive species. In this talk, a mathematical model of the TYC strategy is provided and analyzed. It is shown that the current model may exhibit negative populations that may lead to finite-time blow up in the female population. Modifications to the model to rectify this situation are provided.

Last year we partnered with professors at UMHB's nursing department to develop a new research project. The project targets the Microsoft HoloLens headset as a tool for augmented reality display and data collection. One of the main features requested by nursing professors was a visualization of oxygenated air particles going through the respiratory process. The visualization required a unique solution because it requires at least 100,000 particles to appear accurate and the HoloLens hardware is not capable of performing the calculations in real-time.

We decided to build a custom PC tower that can perform the simulation and then return the data to the HoloLens. This removed the constraints of having to program the engine entirely on a mobile processor. Inside the PC is an RTX Titan graphics card that calculates every particle in parallel by dividing the simulation data between multiple processor blocks. The tower's CPU acts as a communication bridge between the graphics card and the HoloLens, allowing for a real-time augmented reality visual of the particles as they interact in 3D space.

Rather than implementing every aspect of a physics system at once, the simulation is designed to be modular and reprogrammable to perform different operations. One of the major innovations of the engine is that the physics are all calculated on a 3D grid. Each space on the grid contains an updatable vector that affects the properties of the particles as they enter the boundaries. This reduces the number of calculations performed on individual particles. We can currently use the grid to provide a randomized turbulent acceleration or to model a system based on real-world data.

This combination of software and hardware provides an effective simulation engine. The engine will be used as a foundation for visualizing the respiratory process in the next phase of the project.

We will review the established gear graph, \(Gr_n\), and outline proofs for its zero forcing number, \(Z(Gr_n)\), and iteration index, \(I(Gr_n)\). From this we create a new class of graphs, generalized gear graphs, \(Gr^*(m,n)\). We will then outline the proofs for \(Z(Gr^*(m,n))\) and \(I(Gr^*(m,n))\). We then discuss how the proofs for generalized gear graphs relate to hub graphs, a new and larger class of graphs.

We consider a perfect-information, non-cooperative, extensive-form game of finite length, called Ethiopian Dinner. Specifically, we look for optimal strategies, in which the players can expect to do just as well if not better in the game than if they were to employ a different strategy. Research has found that a subgame- perfect Nash equilibrium forms when both players employ what is termed the crossout strategy. We alter this game to include a depreciation of the utility of morsels over time and find that crossout strategy is no longer a subgame-perfect equilibrium. We consider other strategies in this new game form to seek a new optimal strategy.

Draw an even number of points on a plane. Split them into equal groups with a straight line. There are only so many ways to group the points, depending on how they are arranged. Which arrangement(s) can be split the most ways? How do you find this configuration on six dots, and verify it is the best? What about eight? Ten? While it seems straightforward, there are a number of strange and sometimes counterintuitive problems that get in the way. This problem turns out to be a hybrid of combinatorics, discrete geometry, and algebra (depending on how you look at it). Come see (and play along) as we try and find patterns in the points and the lines.

In 1933 George Birkhoff suggested a mathematical measure of aesthetics based on order and complexity. In the years since then, mathematicians, psychologists, and philosophers have tried to refine and extend this measure. We will survey several mathematical measures of aesthetics in images and apply them to identicons.

We review the established Helm graph, \(H_n\), and outline proofs for its zero forcing number, \(Z(H_n)\), and iteration index, \(I(H_n)\). On this basis, we expand the structure of \(H_n\) creating a new class of graphs: Generalized Helm graphs, \(H^*(m,n)\). After deducing \(Z(H^*(m,n))\) and \(I(H^*(m,n))\) using upper and lower bound arguments, we show how Generalized Helm graphs relate to Hubs, a larger and more generalized class of graphs.

Gerrychain is a Python toolset developed by the Metric Geometry and Gerrymandering Group (MGGG) for generating a randomly generated set of district plans of a region, and analyzing the statistical properties of this set. We apply this software to city council districts in Seguin, Texas, and determine the average number of opportunity districts available for minority populations. We then seek to determine a measure of the segregation of a city. In order to evaluate the robustness of our measure, we apply our measure, along with the Gerrychain toolset, to computer-generated cities with varying degrees of racial clustering.

We apply the probabilistic method to sequences of positional games to generate lower bounds for generalizations of Ramsey numbers. We will provide examples of sequences that give rise to classical Ramsey numbers and other well-studied Ramsey numbers.

College is an extremely imperative transition in life, and the push for outstanding grades can seem daunting to some individuals. Nonetheless, almost everyone goes through this challenge, and it's up to us to put in the effort to achieve our desired grade in a class. The most difficult part is figuring out if we are on the right pathway to success. If there is an infallible method that can be used to determine a student's final course average, we could answer that question. This study analyzes aspects linked to success such as quizzes, midterm tests, and attendance using similar theories and techniques as previous studies. R Commander, a statistical software, is utilized to create multiple regression models and scatterplots of the first test, number of absences, and midterm grade versus final average to determine which factors or combination of factors are the most effective in predicting a student's final average. A normal probability plot and residual plot are produced to check the final regression model for any errors. It is recognized that all of these features have an influence on course grades, but midterm average and number of absences together predict final averages most accurately. Since these factors are controllable by the student, this may aid him or her in increasing the likelihood of succeeding in a college class. Additionally, this can help the professor or instructor know in advance how well a student will perform and target any weaknesses early in the course.

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 (MCMC) methods used successfully in the PA Supreme Court case, work to make MCMC more widely available via the Python package GerryChain, and a recent improvement to MCMC called recombination. We will also discuss several commonly used compactness metrics and present a new idea called transit time compactness that aims to measure cohesiveness of people, not just land.

Division I sports are some of the most popular pass times for people around the country, and the schools with successful teams bring in some extreme revenue. There is an ongoing debate on whether or not Division I athletes should be compensated with some of the money their university is making with their name, or if the players should be focusing on their education and not money until the become professional athletes. Using varying models, we will decide if it is feasible for a university to pay their players, and in which way benefits each party the most.

Herb spirals are a type of garden structure usually outlined with stone where edible and medicinal herbs are planted in an ascending spiral helix. Being partially vertical, spiral helixs create multiple surface conditions depending on the cardinal direction and vertical placement of the plant. This paper aims to compare two different spirals: a Fibinnoci and Archemidean and their use in herb spiral gardening.

Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve as the caustic (envelope) of light rays normal to it, and study the Euler's problem for general caustics. The resulting curves are characterized explicitly in two special cases: when the rays are at a constant angle to the curve (skew evolutes) and when they are reflections of parallel rays by it (caustics by reflection). Aside from the analogs of classical solutions known to Euler (logarithmic spirals, cycloids), we encounter infinite-dimensional families of self-similar curves that solve delay differential equations.

The Teichmuller space \(T(F) = \text{Hom}^{dfp}(\pi_1(F),PSL_2(\mathbf{R}))/PSL_2(\mathbf{R})\) is a very complex object. However, a space \(F\) may be decomposed into ``pairs of pants,'' spaces whose interior is homeomorphic to a sphere with three puncture. Given a ``pants decomposition of F,'' one can parametrize \(T(F)\) by pairs \((l_c, t_c)\) describing the lengths of the boundary components of pairs and a certain amount of twisting before gluing the boundaries together. These pairs of pants have a unique hyperbolic structure determined by the lengths of the boundary components. Thus we get a hyperbolic structure on all of \(F\). The presentation is meant to show the elementary properties of hyperbolic space that are relevant to Teichmuller theory. In particular, the equations of geodesics on the upper-half plane model will be derived and the action of \(PSL_2(\mathbf{R})\) on hyperbolic space will be explored.

Determining a minimum distance under a given constraint is always an intriguing question. I will discuss solutions to the following question: A farmer owns a one mile by one mile patch of land that is fenced around its perimeter. He wants to split the land equally between his three daughters. To do this, he will add interior fencing and he wants to use the least amount of fencing possible. How could he do it?

We analyzed the function \(\displaystyle g_n(x) = \sum_{k=1}^{n} \frac{|cos(k\pi x)|}{k}\) at various \(n\) values. The graphs of \(g_n\) have interesting symmetrical properties as well as minimums that are non-differentiable points. In particular we explored dyadic rational points where local minimums occur. We will give some results about when \(g_n\) will share a minimums with \(g_{n+1}\). We will end with some observations and open questions about the graphs of \(g_n\).

Senator Hattie Caraway (D-AR) was the first elected female Senator in United States history. Her tenure in the Senate lasted over twelve years, but the majority of research viewed her as an appendage of fellow Senator Huey Long (D-LA). Senator Caraway is viewed as having traded her vote for reelection support from the popular Huey Long. However, closer analysis of voting data through DW-Nominate scores offers insight into the voting relationship between Senator Caraway and Senator Long. In this presentation, we will examine the DW-Nominate scores of the two senators with differential equations, revealing a far more complex relationship between Caraway and Long. With this data, we argue that Huey Long aided Caraway not to buy her vote but to protect a crucial ally in the Senate.

Zero-order logic consists of the fragment of first-order logic with no quantifier symbols. We prove the completeness theorem for zero-order languages that contain only the predicate symbol of "=" which we call the equality symbol. We conclude with a sketch of the completeness theorem for general first-order logic.

How much do your SAT scores, your scholarships, or any other 'numbers that defines you' actually predict about your likelihood of succeeding as a STEM major. We look at historic data and try to answer this question.

The Apollonian Gasket is a fractal-like object in two space constructed primarily by using Descartes' Theorem. While the Gasket has been studied using geometry and group theory, we will look at the Gasket from a topological viewpoint. We present a method that produces a bar code from a filtration of sub-level, left-level, and sub-radial sets for any nondegenerate Apollonian Gasket given the radii of its three starting circles.

Arrow's impossibility theorem shows that, in the context of linear ballots, a voting system that is independent, Pareto, and has no finite decisive coalitions must be a dictatorship. Assuming the axiom of determinacy, we provide an new proof, of Arrow's theorem in the context of infinitely many voters.

Gonorrhea, caused by the bacterium Neisseria gonorrhoeae, is the second most prevalent bacterial sexually transmitted infection (STI) with 87 million new cases reported worldwide, according to the World Health Organization (WHO). Gonorrhea has developed resistance to multiple treatment therapies within the past century due to the introduction and misuse of antibiotics. The decrease of effective treatment plans develops a posing threat of an untreatable gonorrhea infection and a global health crisis. We aim to study a comprehensive treatment strategy with respect to drug resistance in gonorrhea and its cost-effectiveness. We develop a mathematical model of gonorrhea's resistance with respect to two dual treatments recommended by the WHO, ceftriaxone with azithromycin and cefixime with azithromycin. We also perform a cost-benefit analysis which compares the suggested treatment plans to minimize the emergence of drug resistance. We numerically simulate our model and analyze the reproductive number based on estimated parameters, such that, we conclude the costs of treatments are minimized if more than 50% of individuals are successfully treated.

We examine problems that arise within the field of dynamics and paper folding. Consider two parallel rays, which can be thought of as an infinite strip of paper with parallel sides. One can approximate the angle \(\frac{\pi}{3}\) on this infinite strip using a specific folding algorithm. The same folding algorithm can be applied to strips of paper bordered by different types of curves. For this project, we first consider the system where the folding algorithm is applied between the curves \(y=0\) and \(y = 1 + \frac{\sin{x}}{x}\). As \(x\) approaches infinity, the situation nearly mimics the case of parallel rays, for which it is already known that this folding algorithm will approximate \(\frac{\pi}{3}\). However, do the slight variations in concavity of the top curve have any effect on the angle approximated by the folding algorithm? We offer a solution to this problem, and use this solution to shed light on a generalized variation.

Bacteriophages are a class of viruses that infect and destroy bacteria. For this reason, they are an emerging focus in research due to their potential use in treating antibiotic resistant bacterial infections. Tarleton State University, as part of the international HHMI SEA-PHAGES bacteriophage discovery program, is working on an interdepartmental project between the biology and mathematics departments to create a mathematical model for the complexity of bacteriophage infection of host cells. A system of 4n+1 differential equations were derived to model these interactions. This system of equations was then used to simulate results for the infection outcomes of the different populations. These results were then evaluated in search of equilibria for populations in a given parameter space. It is our hope that these simulated equilibria can be used to identify infection outcomes for bacteriophages used in clinical settings to speed up testing before usage and effectively improve patient outcomes.

In the late 19th century, the Norwegian mathematician Sophus Lie defined one-parameter Lie Groups to find symmetries of differential equations. While it is not a regular part of undergraduate differential equations courses, the Lie Group method is a useful approach for analyzing differential equations of all kinds. Here we review the Lie Group method, including a discussion of key concepts such as the Lie series and one-parameter groups of transformations. We cover the symmetries of well studied equations (e.g: the wave equation) and finish by discussing possible applications of the Lie Group method to our planned experiment.

Matter and energy have a complex relationship in both theoretical and experimental physics. These phenomenon can be described through the mechanics of wave functions and quantum field theory, but are difficult to simulate on larger scales without significant concessions in order to optimize computation. By utilizing common generalizations about the properties of particle systems, as well as techniques such as Fourier decomposition, systems of field disturbances can be modeled while preserving arbitrary levels of accuracy. In practice these models assist with visualization of scaleable stateful systems. For example, the properties of microwaves that allow for them to pass through glass with ease while also getting absorbed by bodies of water. Techniques discussed within this presentation will provide graphical insight into said properties.

We study the generalized graph splines introduced by Gilbert, Tymoczko, and Viel and focus on an attribute known as the Universal Difference Property (UDP). We prove that paths, trees, cycles, and certain theta graphs all satisfy UDP. We also show that UDP is a structural property of edge labeled isomorphic graphs. We use Prüfer domains to illustrate that not every edge labeled graph satisfies UDP.

A two-terminal graph is an undirected graph with two specified target vertices. If each vertex of such a graph fails independently with the same fixed probability (and edges are perfectly reliable), the two-terminal node reliability is the probability that the target vertices are in the same connected component in the induced subgraph of all operational nodes. A two-terminal graph is uniformly most reliable if its node reliability polynomial is greater than or equal to that of all other two-terminal graphs with the same fixed number of vertices, n, and edges, m. In this paper, we show that there is always a uniformly most reliable two-terminal graph. Furthermore, when the distance between the target vertices is restricted to be at least three, we completely classify which values of n and m produce a uniformly most reliable graph and which do not.

In this presentation, we'll model the spread of rumors on a small college campus using the likelihood or ``trust level'' of each person spreading the rumor to another. In my model, the nodes of each graph represent people while the edges represent the probability that person will spread the rumor to them and vice versa. Come watch as the rumors spread across the small campus of Southwestern University where there is a complex web of social connections and trusting just one person with your secret is dangerous.

How many different ways can you express 130 as a sum of four squares, such that \(a^2 + b^2 + c^2 + d^2 = 130\) for \(a,b,c,d\) as non-negative integers? This presentation will focus on how to interpret integer problems using geometry, how to solve efficiently, and why it works.

SET is an addictive game with many hidden mathematical properties still being discovered. We explore the structure, relationships, and geometry of the "SET Universe" consisting of all possible collections of any size from the 81 cards in the game SET. A variation of the game introduces certain arrays of nine cards called "comets." We compute the number of comets and the probability of encountering one in the SET Universe. We then generalize our results to arrays of any number of cards.

The progression of neurodegeneration in Frontotemporal Dementia (FTD) follows from the abnormal accumulation of proteins TDP-43 and/or tau that can interfere with and inhibit the normal interactions in and between neurons in the brain. This process, and how this process can be slowed with treatments, can be modeled by Markov Chains and thus, give possible insight into the signs and symptoms of FTD that correlate with varying levels of protein accumulation.

Mersenne primes, a set of prime numbers of the form \(2^p - 1\), are the subject of much fascination. Only 51 have been found to date, with the largest among these almost 25 million digits long. Existing research has suggested that the distribution of the integers within these prime numbers is uniform. Drawing from work by Benoit Mandelbrot applying chaos to the realm of finance and securities trading, this project uses samples from Mersenne prime M82589933 to build a fictional portfolio of stocks in the S&P 500 index using data from the last 10 years. The prime was split into strings of digits which were translated into ticker symbols. Each week, stocks were either held for the next week or sold and replaced with another prime-generated ticker. The performance of the portfolio was compared with market averages for that decade to determine whether success in the market relies on skill or merely random chance.

In the last redistricting process Texas Congressional District maps were contested at the Supreme Court on the basis of racial gerrymandering, which is drawing political maps to dilute the voting power of groups based on race. Specifically, it was argued that Texas Congressional District 35 was packed with a greater percent of Hispanic/Latinos than that of its surrounding area to dilute their voting power in neighboring districts such as 21, which is less Hispanic/Latino than its surrounding area. To investigate this, we compare the percent of Hispanic/Latinos in each district to that of the district's convex hull, which is the smallest convex region from which the district was drawn. We find that the percent of Hispanic/Latinos in District 35 and two other districts are significantly greater than that of their convex hulls, suggesting that these districts were packed, and that the percent of Hispanic/Latinos in several neighboring districts including 21 are significantly less than that of their convex hulls, suggesting that these districts were cracked. We then explore other approaches to investigate racial gerrymandering.

We will provide an in-depth analysis, using Markov Chain techniques, of a major issue the National Football League is facing: the unfair competitive advantage that is gained by the team that wins the coin toss at the beginning of the overtime period. We will first analyze the old system before the rule changes in 2011 and then the one that is currently in use to see if the advantage still exists. Then we will propose and examine a new overtime system that eliminates the sudden death nature of overtime. We will also look at an alternate set of data values for the variables used in the problem by isolating data strictly from overtime periods to compare the different systems.

Motzkin paths are integer lattice paths that use the steps \(U=(1,1)\), \(L=(1,0)\), and \(D=(1,-1)\) and stay weakly above the line \(y=0\). We generalize Motzkin paths to allow for down steps with multiple slopes and for various coloring schemes on the edges of the resulting paths. These colored, higher-order Motzkin paths provide a general setting where specific coloring schemes yield sets that are in bijection with many well-studied combinatorial objects. We develop bijections between various classes of colored, higher-order Motzkin paths and certain subclasses of \(\ell\)-ary paths, including a generalization of Fine paths, as well as certain subclasses of $\ell$-ary trees. All of this utilizes the language of proper Riordan arrays, and we also include a series of results about the Riordan arrays whose entries enumerate sets of generalized Motzkin paths.

In the Traveling Salesman Problem (TSP), you are given a list of cities and the distances between each pair of cities. The goal is to find the shortest possible route that visits each city exactly once and returns to the original city. The TSP has a wide range of applications in many different industries, including (but not limited to) optimizing mail and shipping routes, guiding industrial machines, mapping genomes, and improving autonomous vehicles. For centuries, traveling salesmen, politicians, and preachers have tackled their own versions of the problem. Within the last century, however, mathematicians have given it more formal thought, and the TSP has become one of the most popular problems in the fields of mathematics and computer science. The time to find an exact solution is often impractically long, which has lead to the development of numerous approximation techniques, ranging from linear programming methods to nature-inspired models. Here, we present a novel N-body approach to the TSP.

Partisan gerrymandering is the act of selectively drawing the borders of an election district in order to manipulate the results of the election to disproportionately favor one party over the other. We studied three different mathematical methods of detecting gerrymandering: the efficiency gap using arithmetic, the Markov chain Monte Carlo method using probability, and the convexity coefficient using geometry. We will discuss the merits and flaws of each of these methods and their mathematical implementation.