Ryan Hausner
Welcome to my mathematics website, where I post papers that I write. This collection consists of research papers, demonstrations of interesting topics, and projects.
Papers
Abstract Algebra, Number Theory, and Proof Writing
- Quotient Groups and Applications: Overview of quotient groups and applications to fields such as cryptography.
- Finite Induction and Binomial Theory: Induction and Binomial Theory, thoughts on David Burton's Elementary Number Theory.
Combinatorics
- Chess and Combinatorics: A proof of why 9 queens can't be placed on an 8×8 board using combinatorics.
Analysis & Calculus
- Construction of the Real Numbers from the Rationals: A summary of how the reals can be constructed from the rationals using Dedekind cuts.
- Proof of Euler’s Identity and Formula: A proof and explanation of Euler's famous identity with some applications.
- Derivative of a Factorial: The derivative of the factorial using the Gamma function.
- Logarithmic Differentiation: A guide on how to take derivatives of functions raised to the power of other functions.
- Non-Elementary Antiderivatives: A collection of non-elementary antiderivatives and how to integrate them.
Numerical Methods
- Numerical Methods: A comparison of modern numerical methods for solving differential equations and their error.
- Phase Space and Model Predictive Control: Overview of phase space, a powerful way to visualize differential equations and a simple model predictive control example.
Linear Algebra and Machine Learning
- Singular Value Decomposition: An introduction and concrete example of Singular Value Decomposition, a foundational machine learning algorithm.
Interdisciplinary
- Mathematics and Music: Exploring some of the ways music can be described using mathematics through pitch, intervals, and overlays.
Research
Analyzing My Classroom: Acoustic Modeling Using the Finite-Difference Time-Domain Method
Abstract: Our senses are one of the most vital parts of how we understand our surroundings. One environment that students and faculty at USC spend most of their time in, is in the classroom. Sound travels as a wave, and can be described mathematically using a partial differential equation, the wave equation. We wanted to understand how sound travels in a USC classroom and compare our understanding in the real world to simulated data, using a numerical method, Finite-Difference Time-Domain (FDTD). This connects the field of acoustics with the field of mathematics. In order to accurately simulate the acoustics of the room, we needed to measure the physical properties of the room, such as the length, width, and height of the space. It is crucial to understand whether sound is going to be reflected or absorbed, and that is dependent on the material of the space. We then used a source at a position in the space and a microphone to record that source, creating an impulse response. We then analyzed these for reverberation times. By acoustically analyzing DMC 157, we found that there are extremely high reverberation times (14-24 seconds),at speaking frequencies (80-300 Hz), where an average classroom should have a reverberation time between 0.6 and 0.7 seconds. This means that the materials in DMC 157 are very reflective, causing copies of the initial sound wave to propagate, causing the initial sound to emanate for longer. We also found that as frequency increases, reverberation time decreases due to higher frequencies being more directional. Using FDTD, we ran a simulation using the numerical scheme and the geometry of the room, by playing a source and receiver in the locations that were in the classroom. The simulation confirmed our results that the reverberation in DMC 157 is very high.
Presented at: University of Southern California Undergraduate Symposium for Creative and Scholarly Work, USC, April 2025.
Calculus Textbook
The intention of this text is to create an intuitive approach to single variable calculus.
Conferences & Presentations
University of Southern California Undergraduate Symposium for Creative and Scholarly Work
Presentation: Acoustic Modeling Sound Propagation in Classrooms Using Finite-Difference Time-Domain
Location: University of Southern California
Date: April 2025
About
My name is Ryan Hausner and I am a current undergraduate student at the University of Southern California in Los Angeles, California. My current degree program is Applied and Computational Mathematics (BS). My research interests include mathematical modeling, numerical methods, and mathematical physics (primarily wave equations).
Contact
Email: r.hausner14@gmail.com
© 2025 Ryan Hausner. All rights reserved.