### Welcome to Mathematics Senior Seminar!

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Senior thesis talks are highlighted in red.

Turnitin information: Course ID 18528800, password carthage.

Senior Research Group Assignments

Everyone needs to be in class for talks at 12:15

Snavely, starting Monday, October 8

Monday

11:45:
Erin

12:15: Sean

12:45: Anna

Friday

11:45:
Gladys

12:15: Benjamin

12:45: Tommy, Juan, and Mary

Yaple

Schedule for research meetings, held in Dr. Yaple’s office

Monday 11:45:

9/7: The entire class will meet today. We will establish the class procedures and expectations, and begin planning the schedule for the remainder of the semester. The exercises assigned today are due on 9/10.

* **PLEASE DO THIS AS SOON AS POSSIBLE - PREFERABLY BEFORE YOU LEAVE CLASS TODAY! **WE NEED THIS INFORMATION FOR PLANNING MONDAY'S CLASS. Take this short thesis progress survey. You will need to log in to your Carthage
account.

**Please complete these tasks before class on Monday, September 10.**

* Create an account on turnitin.com and enroll in our course. The relevant information is in the header of this page.

* Sign up for your free MAA membership at this link.

* Read the course introduction handout carefully. You are responsible for knowing what is in that document!

* Read the three articles provided in the links to the right. Be
prepared to discuss the two papers. Here is one article on giving a good powerpoint presentation, here is the other on a mathematical topic. You will need the login and password given in class. Also, read this article by Francis Su on good mathematical writing.

* Watch this great video on the explaining a concept to people of different ages!

* Be ready to sign up for a time to present your first talk. We will fill out the schedule on Monday.

9/10: We will work on giving a good presentation. Please come ready to discuss the articles assigned on Friday!

9/14: Individual meetings with students. We will not meet as a class, but you will meet individually with your thesis advisor.

9/17: Snavely/Yaple groups meet as a class in one of the math classrooms.

Before Friday's class, submit a copy of a thesis template to turnitin.com. Also, anyone interested in pursuing Honors in Mathematics should contact Drs Yaple and Snavely via e-mail by the end of the week.

9/21: Friday research groups meet. We meet as a class at 12:15 and will hear three talks.

Sean: Families, Ropes, and a Well

Jordan: The Fuel Economy Singularity

Brady: Mathematical Modeling of Surface Water Waves

9/24: Monday research groups meet, with talks at 12:15.

Jiachen: improving the efficiency gap

Gladys: Horsing Around

Mary: Networking Self-Driving Cars

9/28: Friday research groups meet, with talks at 12:15.

Nathan: Optimally Observing Orbiting Orbs

Charles: How should self-driving cars drive

Mike: Inequality for the volume of a Tetrahedron

10/1: Monday research groups meet, with talks at 12:15.

Trevor: The Barn and Pole Paradox

Anna: Non-transitive Dice

Erin: Bingo! What are the odds?

10/5: Friday research groups meet, with talks at 12:15.

Juan: Title

Benjamin: Carcassonne in the Classroom

Tommy: Modeling March Madness

10/8: Monday research groups meet.

10/12: Friday research groups meet.

10/15: Monday research groups meet.

10/19: Friday research groups meet.

10/22: Monday research groups meet.

10/26: THE ENTIRE CLASS NEEDS TO MEET AT 11:45 TODAY TO HEAR THE TALKS. Friday research groups will meet after the talks if there is time.

Mary P:

Mary H:

Lexie:

Jordan:

10/29: No class today due to Fall Break!

11/2: St. Norbert Conference!

Research groups meet as scheduled by your professor. We highly recommend that you go to the conference, especially if you have never attended an undergraduate research conference.

11/5: Monday research groups meet.

11/9: Friday research groups meet.

11/12: Monday research groups meet

11/16: Friday research groups meet

11/19: Monday research groups meet

11/23: No class due to Thanksgiving Break!

11/26: Monday Research Groups Meet

11/30: Friday Research Groups Meet

12/3: Senior Thesis Talks

Anna McHatton:** Heads Up: The Study of Finding the Expected Number of Coin Flips to Absorption**

Abstract: Imagine we have a group of coins on a table in front of us, all facing heads up. We will flip all of the coins once and remove the coins that face tails up. This presentation will discuss how many times we have to repeat this process to remove all of our coins. We will explore and show examples of Markov chains and transition matrices, and discuss how they aided in solving our problem. We will show visuals of our expected number of flips for any number of coins, and compare this graph to one showing the growth rate of the expected number of flips. We will discuss how we found patterns in our results, suggesting the future possibility of finding a function for our problem.

Brady Bresnahan:** Reaction Diffusion Equations and Biochemical Pattern Formation**

Abstract: Many biochemical reactions can be modeled using reaction diffusion equations, some of which form patterns. Reaction diffusion equations are partial differential equations that describe how the concentrations of two species change over time when subjected to diffusion and chemical reaction. These models contain the two-dimensional diffusion equation that is mathematically represented as a second derivative in space. However, the models differ in their terms that describe the interaction of the species. The interaction is described by nonlinear combinations of the dependent variables or concentrations. The two models that we will explore are the activator-substrate reaction diffusion equation and the activator-inhibitor reaction diffusion equation. The reaction diffusion equations will be nondimensionalized, solved, and plotted using Mathematica to see what parameter values will generate interesting concentration patterns at various times.

Charles Gallagher:** The Influence Of Learning Rates On Forward Feeding Neural Networks**

Abstract: In today's society, artificial intelligence and neural networks seem to be everywhere. We see them when we get in our car and turn on Google Maps, we see them when we open up any social media site, and we even see them in our homes with items such as the Alexa that act as smart personal assistants. With this exponential increase of neural networks, we must continue to find ways to optimize these networks or else we risk plateauing our progress due to lack of time. This paper will be studying the effects of increasing and decreasing the learning rate in a feed-forward network as an effort to optimize the network by finding the best-fit learning rate.

Course evaluations are available. Please fill out a course evaluation.

12/7: Senior Thesis Talks

Mary Hussey:** Minimizing Transpositions With Anagrams**

Abstract: In any permutation within a symmetric group, the minimum number of transpositions required to return the permutation to the identity can be algorithmically determined. This consistent property changes as labels of the variables in the permutations are changed to become indistinguishable from other variables of the permutation. We will explore how introducing these indistinguishable variables may reduce the required number of transpositions and how to predict this decrease through the cycle structure and distribution of indistinguishable variables.

Jordan Weathersby:** TBD**

Abstract: Coming Soon

12/10: Senior Thesis Talks

Sean Kilbourne:** Coming Full Circle in the Circle of Powers**

Abstract: This thesis looks at the problem where there is a positive number taken to a power, m^{p} = n, then the sum of the n digits is taken. When is it true that m returns to itself after being taken to a power and the sum of the digits? Numbers taken to the 400thpower were tested. An interesting idea looked at is when only the identity is true for an exponent. One question asked is how do we know that all possible numbers are tested? This answer is given in Lemma 1. Lemma 1 proves the highest number that needs to be tested and when the lemma is false, then any number higher than the tested number will also be false. Next, what is the lowest number that can be tested throughout the powers and still be true? In the research, there upper and lower bounds up to the 400^{th} powers and the groups are divided into groups of 50’s Then, numbers with offsets are tested; offsets are fixed digits that are added to the overall sum of the digits to see when that offset number returns back to the original number. Lastly, what further research can be looked into from the research completed to go deeper into the original problem in this thesis.

Nathan Fiege:** Mathematically Modeling the Josephson Junction**

Abstract: Josephson junctions are superconducting voltage sources which are widely used in precision instruments. This junction and its voltage can be described by a nonlinear differential equation similar to that of the simple pendulum. Here, we derive this model and analyze the junction’s behavior. Differential equations of this type are difficult to understand and solve explicitly, therefore we used numerical analysis, approximations to the nondimensionalized system and periodic arguments to understand this junction’s behavior. This system exhibits two behaviors depending on the ratio between the bias and critical currents, either zero or oscillating voltage. Further modeling of this system could include analyzing specific behaviors including the AC and DC Josephson effects. Understanding the behavior of the junction has had profound impacts in the metrology world (study of measurement) to the extent that we are using this effect to define the fundamental constants of physics and nature.

Erin Freeman:** TBD**

Abstract: Coming Soon

12/12: Our final exam slot is at 10:30 on Wednesday, December 12. We will have senior thesis talks today.

Gladys Montoya:** Infinite Snowman**

Abstract: Have you ever wondered about what we could do with so much snow we receive each year? How about creating a snowman? But not just an ordinary snowman, I’m talking about an infinite snowman! Join me as we determine a formula that will allow us to calculate the area of any snow-friend we want!

Jiachen Liu:** Comparing the growth of internet access worldwide**

Abstract: In the past few decades, more and more people around the world started to have access to internet. The proportion of people who have access to internet has significantly increased about 10 times in the past 18 years. So, in this thesis I will try to evaluate and predict the trend of increasing internet usage in the future for different counties by using mathematical modeling methods. To start with, a differential equation (dx/dt) is needed to represent the rate of change of the proportion x(t) of internet users for any arbitrary population. In particular, I am interested in finding the solutions x(t) for 10 developed countries and 10 developing counties according to real data and the model we set up. Finally, I will make prediction at the end of the thesis about the proportion of people who have access to internet in the future.

Benji Thorson :** The Probability and Statistics of Password Cracking**

Abstract: Passwords are the lock and key of today's society. In order to keep one's personal data safe, a password is sometimes the only barrier between criminals and personal information. In this presentation, we will attempt to answer the question of whether knowing the requirements of a password makes it easier to break in to. By looking at common password requirements, we can combine them together with combinatorics and a statistical data set to discover how many passwords exist, and how many passwords are actually used.

Tommy King:** Success in the Show: Predicting Major League Production from Minor League Track Record**

Abstract: Major League Baseball has recently undergone an unprecedented analytics driven revolution. More so than ever, baseball teams are spending small fortunes on data analysis. To insure the future success of a team, predicting the relative value of minor league players is an important component for long term baseball forecasting. We will compare two different types of regression models and methodologies to determine the best course of action when determining which factors have the highest predictive values when attempting to forecast a baseball player's major league performance from his minor league statistics.

Trevor Wills:** Generalizing the Formula for the Center of Mass in a Dynamic Solid: Fluid Draining**

Abstract: In this presentation, we will be discussing how the center of mass of a dynamic solid changes. Specifically we will analyze a dynamic solid with fluid draining. To better understand this scenario we will define and explain the relevant concepts, including center of mass and revolved solids. With this knowledge we will make simplifications to generalize a formula for location of the center of mass and show its applications.

**Spring Talks**

Day, Time, and Place Forthcoming

Juan Sanchez:** TBD**

Abstract: Coming Soon

Michael Jones:** TBD**

Abstract: Coming Soon