Abstracts presented to the 33 Texas PDE seminar

s presented to the 33 Texas PDE seminar Austin TX April 10, 11 2010 Reza Ahangar Texas A & M UniversityKingsville e-mail: [email protected] Stochastic Modeling of Multistage Carcinogenesis Mutations Oncogene accelerates cell division and tumor suppressor gene slow it down. Both play important role in developing cancer tumor. Initially we accept Hardy-Weinberg assumption which implies that each genotype for tumor suppressor gene T and oncogene O that is TT TOC OO is equally fit. That means the expected number of genes an individual contributes to the gene pool of the next generation is independent of genotype. A stochastic model for the interactions between the tumor suppressor gene T and the oncogene O is developed. The gene pool can be described from one generation to the next by the sequence p(n) where p(n) denotes the frequency of the tumor suppressor gene pool of type T immediately before the n reproduction. We design the stochastic interaction function of multistage mutation based on the formula in our model. In this formulation the tumor suppressor gene has a selective advantage. We assume genotypes have different selective factors and design our interaction between stages using a population genetic model. Giles Auchmuty University of Houston e-mail: [email protected] Orthogonal Bases and Reproducing Kernels for Hilbert Spaces of Harmonic Functions The space of finite energy H1-harmonic functions on a region Ω has a basis of Steklov eigenfunctions that are L2−orthogonal on the boundary and have orthogonal gradients on the region. This leads to a constructive characterization of the boundary trace spaces H1/2(∂Ω) for the region with explicit formulae for inner products and some standard operators. In particular there are general formulae for the Poisson kernel and for the solution operators for Robin and Neumann boundary value problems on the region Ω. This analysis is extended to a 1-parameter family of trace spaces Hs(∂Ω) and their harmonic extensions Hs+1/2(Ω). We shall show that, provided the boundary data is L2, the corresponding spaces of harmonic functions on Ω are reproducing kernel Hilbert spaces with respect to a natural inner product. The reproducing kernels are found explicitly via the Steklov eigenfunctions of the domain. This extends earlier results of Peetre et al and J.L. Lions.