Abstract: The underlying goal of microarray experiements is to identify genetic patterns across different experimental conditions. Genes that are contained in a particular pathway or that respond similarly to experimental conditions should be co-regulated and show similar patterns of expression on a microarray. Using gene networks (or other clustering methods) we can partition the genes of interest into groups or clusters based on measures of similarity. Typically, Euclidean distance or Pearson correlation are used to measure distance (or similarity) before creating a gene network. Both Euclidean distance and Pearson correlation are quite susceptible to outliers, however, an unfortunate characteristic when dealing which microarray data (well known to be quite noisy.)
We propose a robust similarity metric based on Tukey's biweight estimation of multivariate scale and location. The robust metric, the biweight correlation, is simply the correlation obtained from a robust covariance matrix of scale. We provide results demonstrating the robustness and improvement of the correlation method. As well, our method gives an outlier identification procedure which is valuable when dealing with such massive amounts of data.
Abstract: Knots in R 3 which may be parameterized by a single cosine function in each coordinate are called Lissajous knots. We show that twist knots are Lissajous knots if and only if their Alexander polynomials are squares modulo 2. For twist knots which are not Lissajous, we show they are second-order Lissajous, that is, they have Lissajous projections together with height functions given by the sum of two cosine functions. We further prove that all 2-bridge knots and all (3,q)-torus knots have Lissajous projections and give a few general results regarding knots with Lissajous projections.
|Laura Zirbel presenting her work at the AMS-MAA joint meeting.|
|2005 REU participants relaxing at the Pitzer Pool.|