# Senior Capstone Projects

Graduating seniors majoring in Mathematics are required to complete the Capstone course. This is currently offered during the Fall semester. Students are strongly encouraged to that a Capstone Preparation course during the Spring of the Junior year to identify potential topics, investigate background material, and obtain practice with the proposal process.

As part of the Capstone course, each student proposes and carries out an individual research project under the direction of a faculty advisor. Examples of student projects are listed below.

#### Fall 2020

- Fibonacci Sequence Mod m
- The Prisoner's Dilemma's Relation to Toilet Paper During a Pandemic
- How to End Ride Arguments at Disneyland
- Modeling Runs Scored in Baseball
- Optimization of Political Lying
- A Generalization of the Hardy Distribution for Gold Hole Scores
- Modeling Positive Goal-Scoring in Premier League
- A Markov Chain Analysis of XFL Overtime Rules

#### Fall 2019

- Algebraic Curves in Projective Space
- Curves Without Tritangent Planes
- Optimization of Parking at CLU Using Linear Programming
- Optimization of Cal Lutheran Parking Using Optimizations Research
- Analyzing CLU Voting by Shapley-Shubik Power Index
- On the Classification Theorem for Surfaces
- Picks, Bans, and Nash Equilibrium
- Graph-theoretic Theorems in Proofs from THE BOOK
- Analyzing Weighted Bloc Voting Models in the United Nations Security Council using Power Indices
- 9 Ways to Prove the Infinitude of Primes
- Cardinality Bounds on Topological Spaces
- Generalization of the Monty Hall Problem
- An Examination of Hyperbolic Geometry in HyperRogue
- Metrizability in Topology

#### Fall 2018

- Using Stochastic Optimization to Predict Sports Outcomes
- Magic Squares and Elliptic Curves
- Analyzing the Use of Stochastic Optimization through Variations of the News Vendor Problem
- Determining a Model for the California Lutheran University ART Committee Election
- Analyzing California Wildfire Trends Using Bayesian Data Analysis
- Tiling Rectangles and Deficient Rectangles with Polyominoes
- Bayesian Statistics: searching beneath the surface
- Surface Mapping
- RSA and Elliptic Curve Cryptography
- Probability of Playoff Formats
- Mathematical models for the island fox, feral pig, golden eagle system incorporating a strong demographic Allee effect in the island fox

#### Fall 2017

- Predator Prey models
- Optimizing a route at Disneyland
- Bayesian Networks
- Counting Lego stacks
- Analysis of Frieze Patterns
- Knot Invariants

#### Fall 2016

- Micro Economic Theory as a Motivation For Nonlinear Programming
- Bayesian Data Analysis Applied to the Women's SCIAC Swimming Championships
- Applications of Markov Chains in Baseball
- Representations of the Three-Dimensional Rotation Matrices

#### Fall 2015

- Analysis Basic Game Scheduling by TSP and Edge Labeling
- An Overview of Metrization Theorems
- Orbits: A Look at the Symmetry of 4 by 4 Sudoku Puzzles
- Limiting the Size of Topological Spaces Through Cardinal Invariants and Arhangel'skii's Theorem
- An Introduction to Translating Audio Signals into Wavelet Space
- The History of and a Complex Analytic Proof of the Fundamental Theorem of Algebra
- An Analysis fo 2015 Faculty Voting at California Lutheran University
- Finding the Minimum Starting Values of a Sudoku Puzzle that Produces a Unique Solution
- The Pythagorean Triple Tree and Its Many Branches
- Showing the Computational Complexity of Graph Coloring is NP-Complete
- A Mathematical Approach to Analyzing Voting Power with California as 6 States
- How to Find Patterns of Elliptic Solutions Using Modular Arithmetic
- Discovering Hyperbolic Functions Secret Identities through Trigonometric Functions
- Algorithms and Proofs of Three Graph Theoretic Problems
- Entries and Position Impact the Maximum Number of Solutions in a Sudoku Puzzle
- Topological Homogeneity and the Hilbert Cube

#### Fall 2014

- Holling Function Responses: A Look Into Predator-Prey Modeling
- An Array of Cryptosystems and Some Applications
- Influence of Matrix Symmetries on Eigenvalues and Dressing Actions
- Mathematical Modeling of the Hypothalamic-Pituitary-Adrenal Axis
- Duality of Invariants
- A Study of Pascal's Triangle, Square Pyramid, and Tetrahedron Fractal Dimensions and Patterns
- Game Theory and Dominion

#### Fall 2013

- Polynomial Equations and The Fundamental Theorem of Algebra
- Stability Analyses of Several Predator-Prey Models
- Lie Matrix Groups: An In-Depth Look at the Flip Transpose Group
- Linking Markov Chains
- The Dynamics of Newton's Method for Complex Polynomials
- Patterns of Pascal's Triangle and Pyramid with Combinatorics
- Conics and Their Zero Sets
- Elliptic Curves Over Finite Fields

#### Fall 2012

- The Zombie Apocalypse
- The Sprague-Grundy Theorem
- Knot Theory: A Proof of Reidemeister's Theorem
- Lansey's Hypothesis
- Stochastic Optimization and the Newsboy Problem
- Steganography

#### Fall 2011

- Autonomous Robotic Movement Planning
- Burnside's Lemma and Polya's Theorem
- Monopoly and Markov Chains
- The Field of Cryptography Exposed
*p-adic*Analysis, or, The Prime that Killed Calculus- Evolutionary Game Theory
- Investigation into the Mathematical Framework of General Relativity
- Pinpoint: The Colors of Graph Theory and Adjacency Matrices

#### Fall 2010

- Wallpaper Groups- Euclidean and Hyperbolic
- Describing the Pursuit Curve of Missiles en route to Destroy Satellites
- Markov Chain Analysis of NFL Overtime
- Markov Applications to Soccer Formations
- The Hunt for the Classification Theorem for Surfaces

#### Spring 2010

- The Real Line is Uncountable: A Topological Proof
- Riemann Zeta-Functions and Quadratic Fields
- Mathematical Enigmas: Carroll's Guessing Game and a Strategy Puzzle
- Public Key Cryptosystems

#### Spring 2009

- An Investigation of Hyperbolic Triangles
- Being Transcendental: Searching for Transcendental Numbers
- Magic Square Rubik's Cubes
- Explorations of the Four Color Theorem
- A Statistical Analysis of NASCAR Scoring Systems
- Cellular Automata and Group Theory
- Logarithms in High School and Beyond
- Modeling and Predicting Earthquakes with Markov Chains

#### Spring 2008

- The Rank Number of Prism Graphs
- Cracking the Code
- Deal or No Deal: Making Smart Choices Toward Fortune
- Square Patterns in Rayleigh-Bénard Convection
- When Does a Product of Group Elements Equal Its Reverse?
- Shut the Box
- Voting Theory
- A Quantum Mechanical View of the Navier-Stokes Equations

#### Fall 2008

- Quadratic Formula and Beyond! Solving Quadratic and Cubic Equations
- Voting Theory: A Look at Arrow's Theorem

#### Spring 2007

- What are the real numbers and how do we teach them in the middle/high schools?
- The Fibonacci Golf Ball
- Teaching Logarithms
- Schwarzschild Geometry
- Permutation Statistics on Generalized Derangements and Desarrangements

**Fall 2007**

- Meta-Problems in Mathematics
- Magic Squares

#### Spring 2006

- Portfolio Management with the Lagrange Multiplier Technique
- Multi-Strain SIR Model Variations
- Julia Sets Over the Quaternions
- Solutions to Polynomial Equations: Their Role in High School and Beyond
- Potential Orderings of Polyhedra

#### Spring 2005

- Mathematics Behind Patience Sorting
- History and Role of Proofs in Secondary Mathematics Education: a Pedagogical Perspective
- Course Scheduling via Network Flows
- Human Knot
- Is God Rational?
- Analysis of the Current Math Placement Program at CLU
- Blackjack: A Beatable Game
- On the Symmetries of Pascal's Pyramid

#### Spring 2004

- Graph Pegging
- Applying the Hungarian Algorithm to NFL Scheduling
- Voting Blocs in Academic Divisions of CLU: A Mathematical Explanation of Faculty Power
- Mathematical Modeling: Stress Relaxation of Viscoelastic Materials

#### Fall 2004

- Game Theory and Military Scenarios

#### Spring 2003

- Databases for City Planning
- Graph Pebbling
- Number Theoretical Graph Pebbling
- Techniques for Teaching Proofs in High School Geometry
- Twin Primes: A Survey
- Fractals
- Program Scheduling

#### Spring 2002

- Scheduling a Scramble Golf Tournament
- Algebraic Coding Theory
- Shooter's Advantage
- Comparing Craps Table Odds

#### Spring 2001

- What Energy Bar is Best to Fit Your Daily Calorie Needs?
- Earthquakes in California
- Traffic Scenarios
- Soccer Penalty Kicks: Are They Unfair to Goalkeepers?
- Batter Up: A Computer Simulated Look at Baseball
- The Perfect Toss: How to Survive the MCM
- Voting Schemes
- Pin Action!

#### Spring 2000

- Building a Zoom Lens: Matrix Methods in Optics
- Computational Fluid Dynamics
- Title IX and Cal Lutheran Athletics

#### Spring 1999

- When to Say When
- How to Get a Hit
- Salaries of Starting Professional Baseball Pitchers
- Buying a Home as a College Graduate
- Computers, Data, and the Future
- Traffic Control
- The Citric Acid Cycle
- The Tobacco Industry vs. the Health Industry