A bipartite graph G with a bipartite adjacency matrix A can represent a binary matroid, by takin [I A] as the representation of the matroid. This gives a motivation to generalize elements of matroids, connectivity function, branch-width, and matroid minors to vertices of graphs, "cut-rank" function, "rank-width", and "vertex-minors" respectively. Geleen, Gerards, Robertson and Whittle (2003) proved that there is a list of finitely many excluded minors for matroids of branch-width at most k for every k. I proved the same statement for graphs with rank-width and vertex-minor. This shows a new direction for the research of clique-width, which is compatible with rank-width.

Lewis-Sigler Topical Seminar Series 2004, Functional Genomics

A new computational method, Vorticity/Wave Confinement, (V/WC), is described. V/WC has been shown to efficiently treat thin features in complex fluid flow and time domain wave equation problems involving vortices and acoustic pulses. The method allows them to be propagated over arbitrarily long distances with no spreading due to numerical errors. It has also been shown to be effective in representing boundary layers on surfaces as thin vortical layers “immersed” in uniform Cartesian grids. We define these thin vortical or wave regions as “features.” Outside these features, where the flow is irrotational or the wave amplitude vanishes, the method automatically reduces to a conventional finite difference approximation to the correct continuum partial differential equation (pde). The features are treated as a type of weak solution and, within the features, a nonlinear difference equation, as opposed to finite difference approximation, is solved that does not necessarily represent a Taylor expansion discretization of a simple pde. The approach is similar to artificial compression and shock capturing schemes, where conservation laws are satisfied across discontinuities. For propagating features, the result of this conservation is that integral quantities such as total momentum and amplitude, and centroid motion are accurately computed. Effectively, the features are treated as multi-dimensional nonlinear discrete solitary waves that “live” on the computational lattice. These obey a “confinement” relation that is a generalization to multiple dimensions of 1-D discontinuity capturing schemes. A major point is that the method involves a rotationally invariant limiter – a single limiter that is a function of rotationally invariant variables. This is in contrast to conventional discontinuity capturing schemes which may involve a concatenation of separate 1-D limiters, each a function of variables along each axis.

Results will be shown for convection of thin convecting vortex filaments, flow over complex surfaces “immersed” in uniform grids, and thin acoustic pulses scattering from complex surfaces.