## Math senior thesis presentations

Each senior math major has been working on an individual project and will present their thesis research as part of their capstone experience. We would be glad to have you join us in learning about their work and celebrating their accomplishments. Please see below for a partial schedule.

### Friday, Dec. 6: 11:45 a.m. – 1:25 p.m.

**John Ridley ’20, “An Expository Essay on the Math in Blackjack”**To some people, blackjack is just a simple casino card game; however, it is a game that goes beyond simple addition and includes many other facets of mathematics and is rich in statistics and probability. We will explore the different ways mathematics is used in blackjack, identify different variables used in the game today, and we will develop the ideal table that will give the greatest probability to win.

**Mary Phillips ’20, “Representations of Sum Sets in Base n and Ratios Greater than One Half”**Geometric series with ratios r such that 0 < r < 1 take the form 1 + r + r^2 + … = 1/(1-r) when all the terms are added and −1 − r − r^2 − … = −1/(1-r) when all the terms are subtracted. Examining the infinite series with terms (a_n) r^n, where 0 < r < 1 and each a_n is either 1 or −1, we consider a mapping between the sum set of the series with a_n = 1 and a Cantor middle — α set, along with a mapping between the sequences and the base — 1/r representation of the corresponding element in the traditional middle — α Cantor set. We also prove that for 1/2 < r < 1, there does not exist a periodic or eventually periodic sequential representation of zero.

### Monday, Dec. 9: 11:45 a.m. – 1:25 p.m.

**Riley Maguire ’20, “A Complex, Colorful Brain Cell: Quaternion-Valued Neurons”**Here, we investigate neural networks and develop two neurons, one that uses real numbers and one that uses quaternions. Quaternions are 4-dimensional vectors with three complex bases and a real part. In order to make the network classify data as accurately as possible, we minimize a function that determines the inaccuracy, which is called the error function. Since quaternions are non-commutative under multiplication, we determine the most efficient multiplication order for minimizing the error function of the neuron.

**Jake Mitchell ’20, “The Rate of Electric Vehicle Sales in the United States”**In this thesis, we observe the rate of change of sales in all electric and hybrid vehicles, with respect to all vehicles as a whole. In order to do this, we took statistics from several different sources, compared them to one another, and attempted to find the limits of the sales. We wanted to predict when electric/hybrid vehicle sales would outweigh the total sales of vehicles. We use differential equations in order to find these estimates. The differential equation we use creates an “s-shaped” curve that we use to predict the incline and decline rate of sales of hybrid/electric cars, with respect to the sale of all vehicles in the United States. The shockingly low number of electric vehicle sales indicates that electric vehicle sales will pass up overall vehicles in a couple of years. Our predictions indicate that the electric vehicle sale market will not reach 100 percent sales in our lifetime and will begin to taper off in the near future.

**Stefanie Guercio ’20, “Pick A Card, Any Card”**Can a magic trick be done with math? In this paper, you will read how, with a 4x5 matrix, a magician is able to lay out cards with strategy but in a way that looks completely random to the naked eye. This allows the magician to give four different directions to the audience and point them into the correct direction, yet knowing exactly which card each person ends up on. This is done by a professional magician, and we invent a new and original trick with a similar concept.

### Wednesday, Dec. 11: 10:30 a.m. – 12:30 p.m.

**Katherine Turek ’20, “Using Games to Model Semi-Direct Products”**When discussing semi-direct products in abstract algebra, it is often hard to visualize how this operation works. To solve this problem, a game with a series of variations was created to model semi-direct products, specifically based on a group with semi-direct product of order 42. This semi-direct product group contains the dihedral group of order 14 as a subgroup, thus helping students form connections between the different types of non-Abelian groups. Each variation can be adapted to any semi-direct product group, with each variation building on students’ abilities to perform computations in that semi-direct product. These variations are based on games that students have played before such as Candyland, Go Fish, and Rummy in order to increase understanding of semi-direct products.

**David Zikel ’20, “Elliptic Curves and Pairing-Based Cryptography”**In the field of cryptography — in particular, the subfield of elliptic curve cryptography — a recent development is the use of two-input mappings from elliptic curves (or from subgroups thereof) to cyclic groups (often multiplicative subgroups of finite fields), also known as pairings. These elliptic curves are themselves unique, having many properties which would severely reduce their security were they used for conventional elliptic curve cryptography, so the necessary criteria for these elliptic curves (and methods to create curves meeting these criteria) are discussed. In addition, the two pairings most commonly utilized in the field of cryptography, the Weil and Tate pairings, are introduced and discussed, as are efficient algorithms for their computation. An overview of the applications of these pairings to cryptography is given (along with descriptions of currently-published cryptosystems), focusing in particular on two use cases highlighting the differences between conventional and pairing-based elliptic curve cryptography: the use of pairings in the creation of digital signatures, which require less storage space than conventional (elliptic curve) digital signatures ,and the foundational use of pairings in the novel field of identity-based encryption, the problem of creating a public-key cryptosystem in which any given participant’s public key can be determined solely from preexisting public information about that participant’s identity (e.g. the participant’s username or email address).

**Christopher Arbour ’20, “Mathematically Modeling the Effectiveness of Vaccinations and Herd Immunity”**In this paper, we examine vaccinations used to ward against various pathogens through mathematical models. Shown are the benefits of having a “herd immunity” in any given population after being exposed to a pathogen, as compared to the effects of said pathogen on a population lacking access to the same vaccination. By exploring and computing different parameter values in varying models, it is demonstrated that the vaccinated population has a better survival rate than its unvaccinated counterpart, each population being compared to a best-case scenario or control population.

**Cody Phelps ’20, “Modeling a Baseball Player’s Trajectory as a Pursuit Curve”**The idea of pursuit curves was first explored in 1732 by French mathematician Pierre Buogie to describe the trajectory of a pirate ship chasing a merchant vessel. Using this same idea, we make some modifications to explore the path a baseball player would take to catch a fly ball hit along a straight line. We can then compare this path to the measured route efficiency of an MLB player to see how realistic our model is. This pursuit curve involves using differential equations and integrals to solve for an equation that would best describe the curve a baseball player would take to catch a fly ball.

**Joseph Callies ’20, “Tiling a Picture Frame”**The two by n tiling is a simple way to represent the Fibonacci sequence on a piece of paper. In this model, we see a tile taking up two spaces and allows us to visually see the sequence. Here, we want to expand on that sequence by putting the two by n tiling into a square. This research investigates the number of possibilities that a two by n tiling has when put into a two by n square with the middle cut out. We will look at how many tiles need to be placed in order to find out the number of possibilities for the rest of the board.

#### SPONSORING DEPARTMENT, OFFICE, OR ORGANIZATION:

Math

#### FOR MORE INFORMATION, CONTACT:

Dr. Haley Yaple, hyaple@carthage.edu