SENIOR  COMPS PRESENTATIONS

in Mathematics

  

 

Sylow Theorems

Andrew Clarey

For my project I will present the three Sylow Theorems and I will prove the first. The goal is to be able to present the understanding of the theorems by providing examples on the results that come about from the Sylow Theorems regarding any groups and their structure. We will go over a brief background of group theory, followed by the theorems and proofs, and finish with several examples with what we will then know.

 

Constructing Field From Rings

The goal of the project is to discuss ring theory given the knowledge I learned about group theory in Abstract Algebra. I will introduce Groups, Rings, and Fields and discuss definitions and examples of each. Rings and Groups are very similar; however, Groups are only one operation while Rings have two operations. We will discuss ideals in rings, and the idea of a maximal ideal which helps us construct fields. An example of a finite field will be constructed using polynomial rings. I will cover concepts from Saracino’s Abstract Algebra, a First Course sections 17 through 20.

 

Relationships Between Permutations
and Stacking Blocks

Hana Mizuno

Study of Combinatorics is even applicable to middle school geometry homework. In the paper, “Stacking Blocks and Counting Permutations” Lara Pudwell, introduced a new method to find the surface area of unit cubes stacked in a triangular shape as well as an unique set of permutations with forbidden patterns. Pudwell then showed the bijection between the formula of the surface area of the blocks and the permutation method she developed. Additional material of the formula and Pulwell’s permutation method is provided to clarify the analysis by applying materials of Combinatorics.

An Introduction and Analysis
of the Fast Fourier Transform

Thao Nguyen

The continuous Fourier transform converts a function of time x(t) into a function of frequency R(w). While the continuous Fourier transform is used for functions, the Discrete Fourier transform (DFT) processes an array of data points x(t_n) into X(w_k). In this paper we will present the Discrete Fourier Transform (DFT) and the Cooley–Tukey’s original Fast Fourier transform (FFT) algorithm and their time complexities. By implementing these algorithms in Matlab, we present various theoretical examples. Then we apply the FFT to real-world data from research on Occidental’s solar-array and interpret the results.

 

 

 

 

Thursday, December 3rd

4:30pm

Fowler 302

 

**Refreshments Will Be Served**

Everyone is invited!