Recent experiments on the formation of vortex lattices in Bose-Einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns, ones that are free of discrete symmetries and can form in a random environment. We give an overview of an N -particle based Hamiltonian theory which, if formulated in terms of the O(N 2 ) interparticle distances, leads to the analysis of a nonnormal "configuration" matrix whose nullspace structure determines the existence or nonexistence of a lattice. The singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths, in principle, can be classified and calculated by a random-walk scheme which systematically uses the m smallest singular values as a ratchet mechanism to home in on lattices with m-dimensional nullspaces, where 0 < m ≤ N . The resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice, including its size (as measured by the Frobenius or 2-norms), distance between the lattices (hence lattice density), robustness, and Shannon entropy as a quantitative measure of its level of disorder. This article gives an overview of vortex lattice theory from 1957 to the present, highlighting recent experiments in Bose-Einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix. We discuss some of the computational challenges associated with producing large N lattices, the subtleties associated with understanding and exploiting complicated Hamiltonian energy surfaces in high dimensions, and we highlight ten important directions for future research in this area.

A theory capable of producing equilibrium configurations of point vortices in the plane, along with a numerical scheme to compute them, is described. The theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation , where A is the (1/2) N ( N −1)× N non-normal configuration matrix obtained by requiring that all intervortical distances remain fixed, and are the N -vortex strengths. For existence of an equilibrium, A must have a non-trivial nullspace. We consider the singular values of A ; when this has one or more zero singular values, the nullspace of A is non-empty and an equilibrium exists for some choice of Γ . New equilibrium configurations are found numerically by randomly depositing N points in the plane, which generically gives rise to a configuration matrix A with empty nullspace. Using the sum of squares of the k smallest singular values of A as a ‘ratchet’, we ‘thermally fluctuate’ the configuration, allowing each point to execute a random walk in the plane, retaining only those configurations which reduce this quantity at the next step. The configuration is thus driven to one with nullspace ( A )= k >0. These converged states are not necessarily nearby their initial configurations, typically they are asymmetric, and often we can drive the same initial state to several different equilibria. A reverse-ratchet method is also described, which can produce initial conditions that would evolve to a specified equilibrium state. Once a converged final state is achieved, the full singular value decomposition of A is used to calculate an optimal basis set for the nullspace of A and thus all allowable Γ . The distribution of the singular values gives important information on the size of each equilibrium state (as measured by Frobenius norm), their distance from each other (spacing and density) and how far a randomly chosen system of N points in the plane is from the nearest equilibrium configuration with a specified rank, as well as its Shannon entropy.

We consider streamline patterns associated with single and double von Kármán point vortex streets on the surface of a sphere, with and without pole vortices. The full family of streamline patterns are identified and the topological bifurcations from one pattern to another are depicted as a function of latitude and pole strength. The process involves first finding appropriate vortex strengths so that the configuration forms a relative equilibrium, then calculating the angular rotation of the configuration about the center-of-vorticity vector. We next move in a rotating frame of reference so that the configuration is fixed, identify the separatrices in the flowfield and plot the global streamline patterns as a function of the pole strengths and latitudinal positions of the rings. We carry the procedure out for single and double von Kármán vortex streets, with and without pole vortices. The single von Kármán street configurations are comprised of n evenly spaced vortices on each of two rings that symmetrically straddle the equator and are skewed with respect to each other by half a wavelength, while the double von Kármán ring configurations are made up of four rings of n evenly spaced vortices symmetrically straddling the equator.

Our findings, in contrast to previous findings, reveal that panniculectomy combined with pelvic surgery in the morbidly obese is a safe procedure that can decrease the incidence of intraoperative complications and postoperative wound complications without significantly increasing operative time or blood loss.

Aspartate transcarbamoylase catalyzes the second step of de-novo pyrimidine biosynthesis. As malarial parasites lack pyrimidine salvage machinery and rely on de-novo production for growth and proliferation, this pathway is a target for drug discovery. Previously, an apo crystal structure of aspartate transcarbamoylase from Plasmodium falciparum (PfATC) in its T-state has been reported. Here we present crystal structures of PfATC in the liganded R-state as well as in complex with the novel inhibitor, 2,3-napthalenediol, identified by high-throughput screening. Our data shows that 2,3-napthalediol binds in close proximity to the active site, implying an allosteric mechanism of inhibition. Furthermore, we report biophysical characterization of 2,3-napthalenediol. These data provide a promising starting point for structure based drug design targeting PfATC and malarial de-novo pyrimidine biosynthesis.