Mathematics and Computer Science

# 2008 Senior Theses

# 2008 Senior Projects

Spring 2008 | ||

Tuesday, April 1, MCRey 317 | ||

4:00 | McKinzie Fruchtl, Metabolic flux analysis via linear programming | |

4:30 | Zach Person, Modeling with difference and differential equations in neoclassical growth theory | |

5:00 | Henry Phillips, An analysis of degree-6 electrical networks | |

5:30 | Jeanette Reyes, Chromatic polynomials | |

Wednesday, April 2, MCRey 317 | ||

4:00 | Mandy Thomas, Art and computers | |

4:30 | Daniel Levy, Fuzzy Q-learning: An application of fuzzy logic to the Q-learning technique | |

5:00 | David Gould, Fragmentation and recognition of playing card faces |

## Abstracts

Presenter: | McKinzie Fruchtl |

Title: | Metabolic flux analysis via linear programming |

Presentation: | Tue 1 Apr, 4:00pm, MCRey 317 |

Advisor: | Dr. Duff Campbell |

**Abstract:** Metabolic flux analysis (MFA) involves the mathematical analysis of metabolic processes that occur within biological cells. It utilizes a constraints-based approach, where these constraints can limit the behavior of a particular cell. MFA uses optimality principles to predict cellular growth under given conditions. Through the use of linear programming, these optimization problems can be solved. In the event that external constraints become dominant influences on cellular behavior, regulation can be incorporated into the linear programming problem using Boolean logic. The importance of MFA is observed in the engineering industry. It can be used to reduce contaminants in certain compounds, thus reducing costs via time and purification methods.

Presenter: | David Gould |

Title: | Fragmentation and recognition of playing card faces |

Presentation: | Wed 2 Apr, 5:00pm, MCRey 317 |

Advisor: | Dr. Dwayne Collins |

**Abstract:** The use of artificial intelligence in web-based poker clients is dependent upon accurate real-time collection of image data from a poker table window. Current image matching techniques are susceptible to corruption by even minor perturbations of the images. Basic OCR contour analysis techniques are used to examine example images of playing card faces. Card color and image location within the table window are ignored, the primary focus being the relations of contours to their bounding boxes and to one another. The end goal is to find a more robust method of gathering image data, which is resistant to dynamic image changes.

Presenter: | Daniel Levy |

Title: | Fuzzy Q-learning |

Presentation: | Wed 2 Apr, 4:30pm, MCRey 317 |

Advisor: | Dr. Gabe Ferrer |

**Abstract:** Q-learning is a machine learning algorithm which stores predicted reward values for performing actions from states in an N+1-dimensional matrix, one dimension out of N for each mutually-exclusive set of states, plus one dimension for the actions. With finer granularity in states comes an exponential increase in the number of values in the matrix. Fuzzy logic is a mathematical field which extends Boolean logic to real numbers in [0,1]. Presented is an algorithm augmenting Q-learning with fuzzy logic in which the the number of values in the matrix can be reduced, using a coarser granularity per state without decreasing the algorithm's learning potential. This modified Q-learning was tested in the domain of single-player Pong.

Presenter: | Zach Person |

Title: | Modeling with difference and differential equations in neoclassical growth theory |

Presentation: | Tue 1 Apr, 4:30pm, MCRey 317 |

Advisor: | Dr. Dwayne Collins |

**Abstract:** This project will investigate the foundations of Neoclassical Growth Theory and the idea of equilibrium analysis used in interpreting a single output market. The models explored in this project will be derived in two ways. The first will be developed over continuous time. Analysis of this construction involves the use of differential equations. The second and less common derivation utilized discrete time and therefore requires difference equations. Since a fair portion of Neoclassical theory is concerned with how equilibrium exists and how it reacts to certain shocks, influences such as changes in saving and technology will be investigated. The goal is to see if one method will offer anything unique from the other.

Presenter: | Henry Phillips |

Title: | An analysis of degree-6 electrical networks |

Presentation: | Tue 1 Apr, 5:00pm, MCRey 317 |

Advisor: | Dr. Bill Wood |

**Abstract:** In an electrical network there is often the concern of knowing what the effective resistance from the source of the network to the sink. The resistance in a network is a measure of just how poorly the network conducts electricity. This project is an examination of electrical networks where every vertex in the network, except for those on the boundary, has degree 6. It is examined whether or not as the size of the network approaches infinity, if the effective resistance from source to sink approaches infinity, or if it converges to a value.

Presenter: | Jeanette Reyes |

Title: | Chromatic polynomials |

Presentation: | Tue 1 Apr, 5:30pm, MCRey 317 |

Advisor: | Dr. Duff Campbell |

**Abstract:** This project explores the relationship between graphs and polynomials through their respective chromatic polynomials. Graphs with similar characteristics were grouped into families and generated. All graphs up to seven vertices were found by partitioning their total degree sums. All chromatic polynomials were found up to five vertices with the help of P numbers. Relationships were explored between certain family graphs and polynomials in order to generalize the polynomials for a certain number of vertices. All the chromatic polynomials were found to have certain characteristics in common. These characteristic were explored and proved through a series of theorems.

Presenter: | Mandy Thomas |

Title: | Art and computers |

Presentation: | Wed 2 Apr, 4:00pm, MCRey 317 |

Advisor: | Dr. Carl Burch |

**Abstract:** *None*