2021 Mathematics Capstone Symposium
Friday, May 7th
Join the Mathematics Department on Zoom to hear the senior capstone presentations. If you’d like to join the online capstone session, please email Professor Jessica Sklar at email@example.com.
12:30pm – The Gaussian Blur: The Math Behind Image Processing
Much of our world today involves technological advancements that involve complex mathematical concepts. Many of these advancements contain the ability to alter images and videos, much of which is done all from the click of a button. The special image effect that we investigate is called the Gaussian Blur. This specific effect blurs images, or a certain area of an image, in order to reduce detail. The Gaussian Blur uses a type of probabilistic function, called a Gaussian function, in which the effect of the blur itself is dependent on the functions variables. In order to understand how the effect utilizes a Gaussian function, we explore an interesting result when finding the area under a specific Gaussian function, one in its simplest form. The result explains why we must have our Gaussian function divided by the square root of pi. In addition to the Gaussian function, the Gaussian Blur uses a function operation called convolution. This operation is applied to two functions, in which the result of the binary operation is another function. After analyzing the mathematics of the Gaussian function and convolution, we combine these two concepts to analyze how they apply to the Gaussian Blur effect. The Gaussian Blur creates a matrix that contains values that result from using a two dimensional Gaussian function. This matrix is then used to go over each pixel in the image by using discrete two dimensional convolution, resulting in each individually blurred pixel. With such a complicated mathematical process, the Gaussian Blur is quick to blur the desired image. Doing the calculations by hand can be found tedious!
1:00pm – The Gaussian Integers
The study of natural numbers and their simplicity leads to some fascinating problems and comparisons to other number systems. This capstone will focus on the Gaussian integers, their properties, and their analogous nature to integers.
1:30pm – Medical Decision Making: A Game Theoretic Approach
Game theory has many possible games to be analyzed, as well as many different applications of those games. This paper will explore Nash equilibrium in two non-zero-sum games, Chicken and the Prisoner’s Dilemma, and provide additional critiques and benefits of that equilibrium. Then, an example of a medical decision being approached from a game theory perspective will be explored while also considering implications of the previous games.
2:00 – 2:30pm Break
2:30pm – Applications of Euler’s Theorem to the RSA Cryptosystem
The necessity for secure communication has existed for thousands of years, but methods have rapidly changed during the last 50 years. The Rivest, Shamir, and Adleman (RSA) cryptosystem revolutionized secure data transmission by using public-key encryption. In this talk, we introduce the classically used Caesar cipher and compare it to modern RSA encryption. We then discuss contemporary data security and provide an example encryption and decryption of a message using the RSA cryptosystem. Finally, we discuss the mathematics behind RSA cryptography.
3:00pm – Should Linear Algebra be Taught in High School?
In this paper, we will work to answer the question of whether or not high schools should add linear algebra topics into the curriculum in order to provide students with a preparation for college level linear algebra. We will do this by examining two Common Core State Standards for Mathematics, HSA.REI.C.8 and HSA.REI.C.9. Once we develop our base of knowledge by defining terms that will be important to know moving forward, we will go through the most productive ways to teach these concepts in high school and how the skills learned will help in higher level mathematics. All of this information comes together along with other external opinions to help determine whether or not including linear algebra standards in high school helps to prepare students for higher level math courses in college.
3:30pm – The Mathematics of Compound Interests
Compound interest is a powerful thing. Interest is attached to many things in life now a days, like investments, perpetuities, annuities, loans, mortgages, bank accounts, credit cards, insurance products, the list goes on and on. The effects of compound interest can be slow but it has a cumulative way of growing significantly and it can be intriguing how that works. What is happening can be simple, but when we dig deeper into the theory of compound interest and its applications, it can delve into the realm of calculus, where we use integration, geometric series and heavy algebraic manipulation. Compound interest plays a role in the time value of money, and calculating present and future value.
12:30pm – How Multiple Regression and Interactions of Variables Can BE Used to Predict Popularity of a BTS Song
BTS is currently one of the most popular Korean-Pop groups worldwide, with multiple platinum certified hits. Using the Spotify song metrics and other variables we will explore what makes a BTS song popular. We will build and critique prior research by using regression analysis to analyze the interactions between age, music video, and other song variables to learn more about what makes BTS popular. We found correlations between variables and learned the importance of age and presence of music video in a song’s popularity. More research can be done as to what makes a song popular in order to create more successful products in the future.
1:00pm – Exploring the Affect of Sample Size on The Chi Square Statistic
To conduct a Chi-Squared Test, a commonly used convention is that all bins must have a sample size of at least five. This Capstone will explore violations to this rule. Using data collected on the association on drought and deer health conditions, a Monte Carlo Simulation will be conducted. Then a comparison between the theoretical and simulated results of a Chi-Squared test will be made when the sample size is smaller than typical conventions.
1:30pm – Applications of the Jacobian Matrix to Numerical Analysis of Dynamical Systems
Dynamical systems measure state change(s) over time. Whether we are measuring sea level rise due to the warming polar ice caps, or the number of susceptible individuals in a population struggling to confront novel disease, these systems exist all around us. Their interactions lead to complicated dependencies and often chaotic behavior. Numerical analysis provides methods of analysis that enable us to find and describe trajectories, assess stability and equilibrium around critical points, and make inferences about long-term behaviors. One set of methods centers on the use of the Jacobian Matrix, which is a matrix composed of the partial derivatives of the system of interacting dynamical systems. My paper describes a set of these methods that allows for an algorithmic approach to not just finding the roots of interesting system, but also qualitatively describing the system. We will look at the history of chaos theory, and hopefully develop a greater understanding of how, while we are not able to control for chaos, we may be able, through careful analysis, be able to determine how best to use knowledge of trajectories and initial values that optimize outcomes.
2:30pm – Examining the Schrödinger Equation and the Particle in a Box Problem in Multiple Dimensions
The Schrödinger equation is examined under time-dependent and time-independent conditions. First, we provide an introductory background to the field of quantum mechanics and postulates relevant to the scope of this paper. We then derive a solution for the one-dimensional particle in a box. Further, we derive solutions for the three-dimensional particle in a box using a rectangular coordinate system as well as a spherical coordinate system.
3:00pm – Random Number Generation
Random numbers are fundamental to applications in mathematics, cybersecurity, online gaming, etc. Though, how can they be generated? Since the generation of truly random numbers is dependent on natural phenomena, such as radioactive decay or photon entanglement, we look to computers to produce pseudorandom sequences of numbers. Qualities of a favorable pseudorandom number generator are that its numbers are reproducible, yet unpredictable. The first step is obtaining a uniform distribution of numbers, then independent random variable transformations are key to altering these sequences of pseudorandom numbers to contain patterns and possible correlations that are difficult to detect. Although the concept of randomness is not defined, the process of designing algorithms to obtain seemingly random numbers is displayed.
3:30pm – Building and Exploring the Mandelbrot Set: A look into Complex Dynamics
Paul Jean Fischer
The Mandelbrot Set is notable for being arguably the most spectacular depiction of fractal geometry in recent history, but what is it and what makes its intricate structure so entrancing? In this paper we unpack the visually complex nature of the Mandelbrot Set by looking at its iterative process and describe its relationship to dynamical systems. After reading this paper we will have an in-depth understanding of what it takes to build the set and be able to describe it to colleagues.