SENIOR COMPS PRESENTATIONS
in Mathematics
Group
Theory in Physics
Rishi
Bhandia
Group theory is an important subject in theoretical physics, with a wide variety of applications, from particle physics to electricity and magnetism as it allows for the exploitation of symmetries to find solutions to difficult problems. First we will examine group representation theory, and construct multiple group representations for the dihedral group, \( D_3 \). After this we will then present examples of group representations and irreducible and inequivalent representations as well as defining the character of group representations. Finally, the applications of group representations to physical systems will be examined, specifically how they simplify solving certain quantum mechanical problems. Specifically, we will examine the application of group theory to understand how spatial symmetries can be used to characterize solutions to Schrödinger's equation.
Dijkstra's
Algorithm and its Principles
Courtney
Hutton
This talk provides an overview of Dijkstra's Algorithm, a solution to the single shortest path problem. This talk will examine the principles upon which the algorithm was built, namely Bellman's Principle of Optimality. It will also present a proof of correctness for the general form of the algorithm and discuss the time complexity of a naïve implementation of Dijkstra's Algorithm. Variants of the algorithm implementing priority queues with binary and Fibonacci heaps and their respective improvements to the upper-bound time complexity will be examined. Finally, a few current applications of the algorithm will be explored.
Evaluating
the Matrix Exponential
Hector
Lopez
In this talk we will present multiple methods for evaluating the matrix exponential, eAt where A is a nxn matrix of real values. In general, there are many differenr methods that can be used to calculate the matrix exponential, but this talk will primarily focus on three distinct methods that utilize techniques from ordinary differential equations and complex analysis to evaluate it. Putzer's equation, which uses Laplace transforms, the Cauchy Integral Function, and Lagrange Interpolation will be explained and applied to the same square matrix.
Thursday,
March 24th, 2016
4:30-6:30pm (Talks at 5:00, 5:30, 6:00)
Fowler 302
**Refreshments Will Be Served**
Everyone is invited!