Determining Cayley Tables from Cayley Graphs
Wednesday, October 26th
Cayley graphs give a visual representation of the partial structure of a group. A Cayley table represents the complete structure of a group. In a Cayley graph, the elements of a group become vertices and the edges represent right multiplication by the elements of the set S, where S is everything in G except the identity. Using diagrams and tables, I will discuss when is it possible to take the graph of a group and generate complete information (Cayley table) about the structure of that group, from partial information (Cayleygraph), and what characteristics of the given group dictate whether or not this is possible.
Optimizing a Line of Best Fit
Finding maxima and minima of functions is a standard technique taught in every calculus course. In real world situations, however, the functions are frequently too complicated to allow for exact solutions. In this talk I will examine an efficient technique for solving large systems of linear equations. The conjugate gradient method is an iterative approach with strong geometric meaning. I will derive this method and show an example of its use for a large linear system.
Baseball's "Pythagorean Theorem"
November 22, 2011
Sabermetrics is a field of study that engages in scientific research and analyzes several variables that occur on the baseball field. It is used to access a players value in game situations, allowing managers to determine the best combination of players and improving the overall win-loss percentage. With this project, I derived the Pythagorean theorem of baseball which gives an equation to predict the won-loss percentage of a team.
Diophantine Equations in Chemistry
Diophantine equations are equations with integer solutions. Surprisingly, these can be useful for decomposing a complex chemical reaction into individual steps. In this presentation the permanganate/oxalic acid reaction is decomposed into all of the possible individual steps by solving large systems of linear Diophantine equations using integer programming algorithms.
Population Density at the Grizzly Giant
The Grizzly Giant is the most popular sequoia in Yosemite National Park. I will be analyzing data collected to assist the Park's restoration project.
Understanding Multiple Regression Models
Linear regressions are often used to analyze data. During this talk, I will discuss the calculus derivation of fitting the best line, and show that the same results can be yielded using Linear Algebra. After illustrating how the Pseudo Inverse (a Linear Algebra application) can solve over determined systems, I will then discuss how the design matrix is used to help fit data when it contains both qualitative and quantitative information. I will then use various design matrices to analyze real-world data that incorporates the variables lung function (FEV), age, sex, height, and whether or not the person smokes, to better understand the multiple linear regression techniques.
Ever wonder how C3PO, Johnny 5, Robocop, and the Terminator know how to move their body parts? How does their “brain" tell their arms to grab their weapon of choice, or touch the hand of a human being close to them? I will be discussing how mathematics can be used to model such robotic movements. Specifically, I will examine the movement of a robotic arm in a two-‐dimensional plane and use Grobner basis to determine how a robotic arm can reach a particular position. During the talk, Grobner basis will be explained, I will demonstrate how they are used, and how they are calculated. Get ready to explore the fascinating world of robotics and the mathematics involved in making it possible!
The Mathematics Behind Artificial Intelligence
Ever wonder how your brain does stuff? You know walking, talking, or even reading these words. Once upon a time some smart people tried to recreate this “stuff doing" by developing a model based on our brains. It worked so well that today it composes an entire sub-field of artificial intelligence known as neural networks. Come learn about neural networks through a discussion of the evolution of the model from its origins in neuroscience to modern applications and see just how powerful they are.