Posted by: courtneyobrien | January 13, 2009

Mathematical Modeling

I have a number of friends took Mathematical Modeling this past semester.  For their final project, the students were required to develop a model using the mathematical techniques they had learned over the course of the semester. I thought that many of the projects were truly fascinating applications of math to real world problems, especially the ones described below.

Our very own Sandeep modeled how the introduction of a second lane to a one lane circular track could reduce congestion caused by natural, but random, fluctuations in velocity. To model traffic, he used a cellular automata model based on a grid of cells that was updated at each timestep. Each cell was updated according to some rule that was a function of the states surrounding the cell. After a certain amount of time, he then calculated the average velocity of the cars around the track to determine the effect lane changing had on congestion.

Duncan sought to identify the optimal utilization of non-directed donor kidneys in kidney transaction clearinghouses. Clearinghouses match up donor-patient pairs in order to facilitate kidney donations among family and friends.  For example, if a type A donor can’t donate to his friend, a type B patient, a clearinghouse can match them up with a type B donor who wants to donate to a type A patient.  Occasionally, a “non-directed donor” or “Good Samaritan Donor” will willingly donate his kidney to a clearinghouse.  In this case, there are more donors than patients.  Duncan proposed that rather than using the leftover kidney today, clearinghouses could save the extra kidney in order to catalyze more transactions in the future.  Duncan’s model attempted to identify the optimal blood type of that leftover kidney in order to maximize present and future transactions.  He based his model on the integer programming problem by defining compatibility and assignment matrices C and X, where each element Cij and Xij in C and X=1 if the kidneys are compatible and 0 if they are not.

Ricky developed a NCAA College Basketball ranking model using markov chains weighted according to the point margin of individual games.  His model also more heavily weighted games later in the season.

Parth developed a model for analyzing color heterogeneity in films.  He looked at three measures: 1) comparison of each pixel in a frame to adjacent pixels 2) comparison of each pixel in a frame to a grid of pixels evenly spaced across the rest of the frame and 3) comparison of each pixel to the same pixel in the 2 frames before and 2 frames after it.
He found that factor 1 turned out to be nearly useless, and assigned it very little weight to it.  However, the other two factors were very useful.  He also compared his heterogeneity index to average ratings found on IMDB to see if there was any correlation.  He found that animated movies tend to be a lot more dynamic than live action movies.

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