Partisan gerrymandering is the manipulation of district boundaries in order to establish an unfair political advantage for a particular party or group. The goal of this research project is to develop code to evaluate the current districting of a state using computational mathematics. We developed a random walk model using Wolfram Mathematica that generates thousands of district plans. The new districts continue to meet criteria regarding population bounds and precinct adjacencies. These plans are then analyzed to find normal distributions. In order to validate the algorithm, a model state, “Pentucky,” made up of 6 districts and 100 precincts, was used. Two disparate initial district plans were used as inputs for the algorithm; a consistent winning-party average resulted from both district plans. This concept is being further applied with the 3,692 precincts within Kentucky. Using registration statistics, the algorithm will generate a distribution of districting plans with hopes to guarantee equal voter representation.

Euler’s totient function ϕ(n) is defined as the number of integers less than n which are coprime to n. Carmichael’s Totient Conjecture states that for each positive integer n there exists a different integer m such that ϕ(m) = ϕ(n). While investigating a condition posed by Pomerance as sufficient to disprove the conjecture, we discovered bounds for Dirichlet’s theorem of arithmetic progressions given by the Jacobsthal function by reducing its calculation to a deceptively difficult game of modular whack-a-mole. Also, the conjecture would imply that any integer satisfying Pomerance’s condition must be divisible by the square of every prime, hence Jacobsthal’s Ladder.

Using data on 44 high school bowlers from the Eastern Cincinnati Conference, we examine statistical properties of 10-pin bowling scores. We look for a relationship between average scores and standard deviation by using the Wilcoxon Rank-Sum Test and the Rank Correlation Test. Additionally, we determine if bowling scores are normally distributed using Skew-Kurtosis Tests. Finally, we examine Handicap Systems and their effectiveness and impact on high school bowling.

On an n × n chessboard, every two squares are at most two queen moves (or rook moves) from each other. We ask how many pawns need to be placed on the board to increase the maximum possible distance from 2 to some desired number d. We produce an algorithm to answer the question for given values of n and d. For some small values of d, we extend the results produced by that algorithm to all n.

In this presentation, I will illustrate how my students and I have used a problem-posing framework and GeoGebra to pose and solve Varignon problems using four main strategies: Specializing, generalizing, extending, and reversing. To enrich the students’ experience, I start the investigation with the following version of the Varignon’s problem: Let E, F, G, and H be the midpoints of the consecutive sides of a parallelogram ABCD. What type of quadrilateral is EFGH?

Representation theory is the idea that we can learn something new about an algebraic structure, say a group, by realizing its elements as something more concrete, such as matrices or symmetries. Combinatorial representation theory is the idea that objects with interesting discrete characteristics often accompany concrete representations of algebraic structures. Our goal is to exhibit a really nice example of these ideas that we discovered recently in our work with representations of symmetric groups and certain matrix algebras — this is where the numbers-in-boxes stuff will happen.

Knots can be considered as thick-knotted tubes or a stiff thin curves. These considerations influence the shape the knots will have. We discuss the mathematics behind these ideas.

We view the solid angle as a cone spanned by extreme rays. The measure of the solid angle is the proportion of space that the cone occupies. Of particular interest is the normalized measure of a simplicial solid angle. It is well known that in dimension two, the angle between two vectors can be computed using the dot product. In dimension three, the normalized measure of a simplicial solid angle can be obtained using the scalar triple product, as noted by Oosterom and Strackee (1983). In 2006, Jason Ribando discovered that the normalized measure of a simplicial solid angle in higher dimensions can be obtained via a multivariable hypergeometric series, when the series converges. In this talk, I will showcase Python code that I developed in collaboration with Dr. Yuan Zhou, which uses a truncated form of Ribando’s formula to approximate the normalized measure of solid angles.

We will explore aspects of the networked-numbers game—a game played on a graph whose moves consist of “firing” weighted nodes. The networked-numbers game is relatively simple to understand, but its structure is rich with results in combinatorics, root systems, and Lie Algebra. First, we will look at the construction of this game. Next, we will examine the game played on a general Dn-type graph. Lastly, we will give an overview of the significance of this game in algebraic combinatorics.

We will investigate the bounded factorization property of the monoid domain F[X;M], where M is an additive submonoid of the nonnegative rational numbers. We accomplish this task by examining sufficient and necessary factorization properties of the associated monoid M and, in particular, its irreducibles. We divide our work into two sections: we first present a useful characterization of all bounded factorization monoid domains by precisely determining the necessary factorization properties of its associated Puiseux monoid; then, we consider those bounded factorization domains with the finite factorization property.

We describe several ways of how a mathematical knot can curve when drawn on paper, including braid diagrams and spiral diagrams. Using these, we can define knots as “curly,” or as having a braid index greater than its spiral index. From there, groups and families of curly knots can be described and proven to be curly.

Inspired by recent year’s Joint Mathematics Meetings’ Mathematical Art Galleries, our research seeks to produce esthetically pleasing surfaces using Mathematica as the medium. In our work, we used minimal surfaces, surfaces of least area, and tube surfaces, surfaces built around curves. We used code to modify these sets of equations, which were then combined with built-in functions generating a range of results. Our results included 3D prints, movies, and graphics in Mathematica. The 3D prints helped us visually display the position, in space, of the surfaces we created on a real-life scale. We used digitally generated movies to show how a minimal surface can morph into another surface by altering a parameter value. Finally, we altered parametric equations to produce an array of surfaces and tubes to show how minor changes in these equations affect the appearance of each graphic.

Stock options can be a useful tool in any investor’s portfolio. They allow the skilled investor to increase their leverage and possibility for a higher payout with less risk. However, they are only useful if the investor knows how to use them, and if they accurately reflect the price of the option. This presentation hopes to offer some insight in the binomial method of options pricing along with new adjustments to the model in hopes of reflecting a more accurate options price. This, along with the Monte Carlo simulation presented, will hopefully offer some insight into the mathematical investor’s toolbox.

In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, etc. Suppose we have a piezoelectric beam and want to find the observation time of a given signal. To represent this mathematically, we can discretize the spatial variable in a system of coupled wave equations via finite differences. We find that for this particular discretization, the observability time approaches infinity which does not mimic the continuous case, and is unsuitable in terms of controlling the system. To bypass this issue, a filtering technique known as direct filtering is applied, and a (non-optimal) observation time is obtained.

COVID-19 is the virus that is responsible for the pandemic that currently effects the entire globe, causing mass shutdowns and economic disruption. In the United States, over 515,000 people have died due to COVID-19. This analysis began with first analyzing the state with the highest death toll and working to build several models using multi-regression to predict future deaths in this state. After these models were complete, work began on building models for the entire United State in order to predict deaths for the nation as a whole rather than just a singular state using what was learned previously.

Cryptography plays an essential role in daily communication through technology; programming is a valuable skill because of its numerous applications in STEM. Our goal is to develop a course to forge student proficiency in both cryptography and the basics of the Python programming language. In this presentation, we will elaborate on the objectives of the course and how they are to be achieved. Student learner outcomes include proficiency in Python programming, expounding on security properties in cryptographic theories, analyzing and applying cryptography in Python, and considering the corresponding limitations and applicability. The presentation will conclude with an update on the current implementation of the curriculum.

This presentation examines the dynamics of COVID-19 and the factors which affect outcomes within a population. A Susceptible-Exposed-Infected-Recovered compartmental model was developed to represent the various disease-states of a population. Euler’s Method was used to find approximate solutions of the first-order differential equations governing the compartmental model. The model was applied to a population approximately the size of Louisville, KY, in order to simulate a possible outcome of events for the largest metropolitan area in the state. Three scenarios were addressed using the developed model: one resembling the historical course of the outbreak, one in which a mask mandate was applied early in the course of the outbreak but individuals still chose to gather during the holiday season, and one in which a mask mandate was applied early in the course of the outbreak but individual refrained from gathering. Outcomes suggest that significant morbidity and mortality is expected except when masking is applied early on in the outbreak and individuals refrained from gathering during holidays.

The curvature function of a 2D curve describes the shape of the curve regardless of how it is moved or rotated. In music, a motif is string of notes that appears throughout a piece, although sometimes moved up or down. The measure of curvature is a number, and we can assign each note in a motif a numerical value that describes its distance from some fundamental frequency. Because each curve has a unique curvature function, we are able to create a unique song based on a curve, and vice versa. However, this does not easily allow for rhythm. A 3D curve has curvature and torsion, which measure how much it “bends” and how much it “twists”, respectively. We can make curvature rhythm, and torsion pitch, and get a unique song based on any space curve, and vice versa.

We generalize the results of the alternating Dirichlet series first examined by Junesung Choi. Specifically, we examine its analytical continuation and its special values. We also connect values of the Choi Dirichlet series to values of another Dirichlet series. Finally, we connect this last class of Dirichlet series to Dirichlet L-functions, ultimately obtaining values of the alternating Dirichlet series.

In this talk I will announce my favorite YouTube channels which focus on math topics. The title betrays my #1 .

In control theory, a system is exactly observable if, when you measure a certain system property for a finite time, you are able to distinguish any two different initial states on a string. With partial differential equations (PDEs), like the one-dimensional wave equation, there are infinitely many eigenvalues, which makes the PDE infinite-dimensional, so exact observability of vibrations on a string is an infinite-dimensional problem. In practice, however, system properties are observed by the sensors, and sensors can only observe finitely many vibrational modes. Therefore, sensor design for the observed property has to be done for a numerical approximation of the PDE, which makes it a finite dimensional system. However, exact observability does not hold for well-known numerical approximation techniques like Finite Difference Method (FDM) and Finite Element Method (FEM). The major issue we encounter in trying to show the exact observability of the approximated system is that the uniform gap property of eigenvalues is lost, which is not the case for the infinite-dimensional system. This uniform gap ensures exact vibrational observations by the sensors. Without it, sensors cannot distinguish one vibrational mode from another. This happens because the deficiencies of the numerical techniques cause artificial, high-frequency eigenvalues, which results in the uniform gap approaching zero as the mesh parameter goes to zero. This talk will focus on the lack of observability for the FEM space-approximation of the wave equation and a remedy, the so-called “direct filtering technique,” which enables us to prove exact observability. The proofs are based on the so-called “discrete multipliers technique.”

This presentation explores divisibility, periodicity and prime outputs in the Tribonacci sequence in addition to palatable number tricks associated with the sequence. In contrast to the companion Fibonacci sequence which embodies a rich history of productive research, the OEIS (The On Line Encyclopedia of Integer Sequences) indicates the study of the Tribonacci sequence is ripe for discovery. For example, only fourteen prime entries were known as of 2000. Recently I secured the fifteenth and sixteenth prime outputs using Mathematica 12.0 which I will share and was able to completely factor the initial three hundred twenty-five terms. While every integer n enters the Fibonacci sequence no later than the n² term, no power of three beyond nine serves as a factor of any term in our sequence. The Pisano periods of the Tribonacci sequence are messy at best in contrast to the Fibonacci sequence, but illustrate the power of modular arithmetic. This presentation is accessible for undergraduate students who enjoy topics in number theory, discrete mathematics and computer science as well as recreational mathematics enthusiasts.