Mathematics Projects 2009

Linear and Affine Transformation

Faculty Mentor - Dr. Charles Lam

Linear and Affine Transformations are used in computer graphics to help create art and games. Contraction reduces the size of a figure and dilation increases the size of a figure. A rotation is a transformation in which it rotates around the origin. Reflection of vectors across the x and the y axis. Affine transformation is a composition that is composed of a matrix transformation followed by a translation.

The Coloring Problems

Faculty Mentor - Dr. Charles Lam

The coloring problems are such that they are easily understood but require the best efforts to be solved, particularly the four color problem. The concept is quite simple: the objective is to color a map of regions so that no two adjacent regions result in the same color. The task is to find the smallest number of unique colors that will accomplish this goal. Using the process of induction, proving this true for six and five colors is fairly simple, when compared to the four color proof. Francis W. Guthrie first posed this problem in 1852 to his brother when he asked if it is possible to color any map with contiguous regions using only four colors in such a way that no two adjacent regions (regions that share an edge) will be of the same color. The coloring problem has still not been satisfactorily solved. Mathematicians began working on the problem about 20 years after it was posed. They have successfully proved the six and the five color problems through mathematical induction, but the only way to solve the problem as of yet is through exhaustion, which many mathematicians find unsatisfactory.

Statistical Analysis: Connections between valley fever and PM₁₀ data of Tulare County

Faculty Mentor - Jorge Talamantes

Coccidioidomycosis (valley fever) is an infection acquired when the host inhales the spores of the fungus Coccidioides immitis or Coccidioides posadasii. The fungus grows in pristine (uncultivated) soils, but its spores can become airborne and thus infect a host. The fungus spores measure 2-5 micrometers. Thus, it makes sense to assume that if climatic conditions are such that there is a lot of dust in the air, then the same conditions must favor high concentrations of spores in the atmosphere. We test this hypothesis by using the number of valley fever cases reported in Tulare County from January 1996 to December 2006, and looking for statistical correlations with same-time and antecedent atmospheric PM₁₀ concentrations. (This refers to the concentration of particular matter of size less than 10 micrometers.) Our results suggest that, in agreement with similar studies in Kern County (but in contrast with work with Arizona data), a direct dust-valley fever correlation is not statistically significant.