Friday, Nov. 18
Mathematicians have long used power series to represent important functions that arise in mathematics, physic and chemistry. This talk will present a method for constructing an approximating function, namely the Chebyshev polynomials of the first kind.
The Chebyshev polynomials will be constructed using the Gram Schmidt process; a process that takes a non-orthogonal polynomial and creates a polynomial that is orthogonal. Two important theorems will be proved about orthogonal polynomials:
* The Three-Term Recurrence Relations Theorem
* The Differential Equations Theorem
The talk will compare the accuracy of the Taylor series expansion vs. the Chebyshev polynomials for given functions.
Singular Value Decomposition
Monday, Nov. 21
Statistics usually requires a smooth approximation, or an interpolation, for a given set of planar data points to create a modeling function. With techniques such as least squares, higher-order polynomials, or cubic splines we are able to construct a function that models a given set of data points. This presentation focuses on a technique called Singular Value Decomposition to handle surface fitting techniques for three-dimensional data points.
Solving Differential Equations using Matrix Exponentiation
Monday, Nov. 28
Solving differential equations is a common task in mathematics. However, solving a system of differential equations is an application that can become a burden. Linear algebra is the tool that mathematicians use to solve systems of equations and linear algebra can be used to solve systems of differential equations. The matrix exponential shortens and simplifies this task, giving almost a shortcut to solving the system through extensive linear algebra. This talk will discuss the matrix exponential and the use of the matrix exponential to solve differential equations of the form x'=A x, in particular where A is a diagonalizable matrix.
Wednesday, Nov. 30
The Sylow Theorems can be used to find a significant amount of information about finite Abelian groups, such as the number of subgroups and their corresponding orders, only from knowing the order of the group under investigation. This talk will introduce the Sylow theorems, including a proof of Sylow's First Theorem, and the application of these theorems.
Cyclic groups and the generation of the La Loubere Magic Square
Friday, Dec. 2
An odd-order magic square is a square array of numbers in which the sum of any row or any column is the same (including the sum along the diagonals). This talk will discuss the La Loubère Magic Squares and the group theory, number theory, and Fibonacci number properties contained therein.
Monday, Dec. 5
This talk will introduce the properties of Fourier series and Fourier transforms and their application in data processing. We will show how data gathered from an NMR spectroscopy instrument can be processed using Fourier transforms.
Predicting the Position of the International Space Station
Wednesday, Dec. 7
Modeling Competitive Species
Friday, Dec. 9
This talk will discuss the standard competitive species model, which has the usual quality that one of two competing species almost always becomes extinct. This is the usual case with the exception of when the species' population reaches equilibrium. We will investigate these equilibrium points using the techniques of phase-plane analysis and linearization theory. From these techniques we are able to describe the behavior of the populations as they approach the equilibrium point. A final example will investigate the introduction of a predator into the population model for two species.