Symmetric Subgroup Actions on Isotropic Grassmannians

Let G be the group preserving a nondegenerate sesquilinear form on a vector space V, and H a symmetric subgroup of G of the type G1 x G2. We explicitly parameterize the H-orbits in the Grassmannian of r-dimensional isotropic subspaces of V by a complete set of H-invariants. We describe the Bruhat order in terms of the majorization relationship over a diagram of these H-invariants. The inclusion order, the stabilizer, the orbit dimension, the open H-orbits, the decompositions of an H orbit into H\cap G_0 and H_0 orbits are also explicitly described.Comment: 30 page

SED, a normalization free method for DNA microarray data analysis

BACKGROUND: Analysis of DNA microarray data usually begins with a normalization step where intensities of different arrays are adjusted to the same scale so that the intensity levels from different arrays can be compared with one other. Both simple total array intensity-based as well as more complex "local intensity level" dependent normalization methods have been developed, some of which are widely used. Much less developed methods for microarray data analysis include those that bypass the normalization step and therefore yield results that are not confounded by potential normalization errors. RESULTS: Instead of focusing on the raw intensity levels, we developed a new method for microarray data analysis that maps each gene's expression intensity level to a high dimensional space of SEDs (Signs of Expression Difference), the signs of the expression intensity difference between a given gene and every other gene on the array. Since SED are unchanged under any monotonic transformation of intensity levels, the SED based method is normalization free. When tested on a multi-class tumor classification problem, simple Naive Bayes and Nearest Neighbor methods using the SED approach gave results comparable with normalized intensity-based algorithms. Furthermore, a high percentage of classifiers based on a single gene's SED gave good classification results, suggesting that SED does capture essential information from the intensity levels. CONCLUSION: The results of testing this new method on multi-class tumor classification problems suggests that the SED-based, normalization-free method of microarray data analysis is feasible and promising

On Kostant's partial order on hyperbolic elements

We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements.Comment: 7 page

Extensions of Yamamoto-Nayak's Theorem

A result of Nayak asserts that $\underset{m\to \infty}\lim |A^m|^{1/m}$ exists for each $n\times n$ complex matrix $A$, where $|A| = (A^*A)^{1/2}$, and the limit is given in terms of the spectral decomposition. We extend the result of Nayak, namely, we prove that the limit of $\underset{m\to \infty}\lim |BA^mC|^{1/m}$ exists for any $n\times n$ complex matrices $A$, $B$, and $C$ where $B$ and $C$ are nonsingular; the limit is obtained and is independent of $B$. We then provide generalization in the context of real semisimple Lie groups.Comment: 14 page