Grassmannian flows and applications to nonlinear partial differential equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure

Mesostructured Block Copolymer Nanoparticles: Versatile Templates for Hybrid Inorganic/Organic Nanostructures

We present a versatile strategy to prepare a range of nanostructured poly(styrene)-block-poly(2-vinyl pyridine) copolymer particles with tunable interior morphology and controlled size by a simple solvent exchange procedure. A key feature of this strategy is the use of functional block copolymers incorporating reactive pyridyl moieties which allow the absorption of metal salts and other inorganic precursors to be directed. Upon reduction of the metal salts, well-defined hybrid metal nanoparticle arrays could be prepared, whereas the use of oxide precursors followed by calcination permits the synthesis of silica and titania particles. In both cases, ordered morphologies templated by the original block copolymer domains were obtained

On search games that include ambush

We present a stochastic game that models ambush/search in a finite region $Q$ which has area but no other structure. The searcher can search a unit area of $Q$ in unit time or adopt an “ambush” mode for a certain period. The searcher “captures” the hider when the searched region contains the hider's location or if the hider moves while the searcher is in ambush mode. The payoff in this zero sum game is the capture time. Our game is motivated by the (still unsolved) princess and monster game on a star graph with a large number of leaves