Many of our insights into the dynamics of specific nonlinear systems, either in the form of maps or differential equations, have come about through numerical simulations. I would also argue that these simulations are what have led to the general acceptance of the ubiquity of complicated phenomena such as chaotic dynamics and strange attractors by scientists and engineers. However, there are two obvious problems with numerical simulations: (1) they contain errors and (2) they can only be done over a finite time interval. The purpose of this talk is to demonstrate these problems can be overcome by combining ideas from Algebraic Topology with standard numerical methods, without necessarily dramatically increasing the cost of computation.

Cerebral aneurysms are the leading cause of hemorrhagic strokes. In unruptured aneurysms, current treatment risks can exceed the risk of natural rupture. A better understanding of the processes leading to rupture is needed to better select only those aneurysms which place themselves in a high-risk category. Recently, several studies have used image-based computational fluid dynamics to analyze the hemodynamic behavior in cerebral aneurysms. These studies suggest that complex/unstable flow patterns, with concentrated inflow jets impacting on a small region of the aneurysm, have a higher rupture risk. However, one of the limitations of these studies is that they assumed that the shape, and thus the hemodynamic profile, is not dramatically changed by the aneurysm rupture. This case presents a computational analysis of a fatal aneurysm of the basilar artery that was imaged with 3D rotational angiography hours before its rupture, and before the patient could be treated. Cases like this are extremely difficult to collect since unruptured aneurysms are never followed up until they rupture. The goal of this study serves several purposes: to verify that the hemodynamic characteristics derived from patient-specific models indeed place this aneurysm in the high risk category, to better understand the nature of potentially lethal aneurysms, and to offer a solution regarding the nature of the aneurysm’s initiation.

In space plasmas thin electric current sheets are observed to form near the surface of the sun, in the solar wind, and in Earth's magnetotail. The basic physics of this system can be approximated by a diffusion model with a diffusivity that has hysteresis and switches between low and high, depending on local current amplitude. We show that under constant energy input conditions, this system evolves to a state with characteristics of self-organized criticality.

The Schrodinger equation was solved using the Augmented Plane Wave (APW) method for transition metals in both the body centered cubic (bbc) and face centered cubic (fcc) structures. This method predicts accurately the equilibrium lattice parameter and the ground state of all transition metals. The Rigid Band Model tests based on Ruthenium (Ru) and Rhodium (Rh) was then applied to the data to predict the density of states (DOS) at the Fermi level for the rest of the transition metals. This test agreed with direct calculations quite well with only a few exceptions in the hexagonal structures. The APW method was also applied using the virtual crystal approximation to obtain the DOS of binary alloys. The results will be compared with direct calculations of ordered structures.