Analytical Behaviour of Solutions of Boundary Integral Equations for a Nonsmooth Region

The Dirichlet problem for Helmholtz's equation in a domain exterior to some bounded smooth boundary in two dimensionsmay be solved by means of a combined potential of the singleand double layers. In this paper, the problem arising from allowingcorner points on the boundary is investigated. The resultingnoncompact operator is effectively split into singular and compactparts. By using the Mellin transforms, the equation can be convertedinto some Cauchy-type singular integral equations. Consequently,the singular form of the solution is found in terms of r^ßat a corner with 0>ß>1. As a first step towarddeveloping new numerical methods for the problem, one typicalexample is presented to demonstrate the slow convergence ofexisting methods without any modifications. Then the mesh-gradingtechnique designed for singular equations is successfully implementedto restore the order of convergence.     

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficientnumerical methods for the solution of boundary integral equationsdefined on a non-smooth boundary. In particular the solutionof the Helmholtz equation in the exterior of a closed wedgeis studied. The analytical behaviour of the solution of theresulting boundary integral equation (with a non-compact operator)near the wedge is investigated. Numerical analysis of the collocationand iterated collocation method for the problem is presented.Graded meshes are used to reflect the ‘singular’behaviour of the analytical solution, as well as the degreeof the polynomial approximant, in order to yield results with‘optimal convergence rates’. Finally the convergenceanalysis of some modified two-grid iterative methods for thefast solution of the resulting linear systems is given and numericalresults are presented which agree with the theoretical predictions.     

Chain conditions for C*-algebras coming from Hilbert C*-modules

In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.     

POLY-L-LYSINE-GRAFT-PEG COMB-TYPE POLYCATION COPOLYMERS FOR GENE DELIVERY

ABSTRACT

Polycations have been used for gene delivery in vitro quite successfully, however, in vivo applications suffered from serum effects that lower the overall gene drug efficiency. PEG polymers have been used extensively to minimize serum effects and create “stealth liposomes”, biocompatible materials, and proteins with extended circulation. Here, we report our efforts towards creating “stealth polyplexes”. A comb-type polycation, poly-L-lysine-graft-PEG copolymers were successfully prepared by ring opening of PEG-epoxide with e-amino lysine groups of linear poly-L-lysine. The ratios of PEG-epoxide to poly-L-lysine, PEG-epoxide size (2K, 3K, and 5K), and poly-L-lysine size (10K, 26K, and 38K) were varied. Copolymers with as little as 2% grafted PEG chains sterically stabilized DNA/copolymer complexes (polyplexes) even at charge neutrality. These polyplexes, formed with copolymers with various size of PEG chains grafted on various lenghths of poly-L-lysine backbone, remained relatively small, approximately 100 nm in saline With higher degree of grafting, the binding was significantly diminished. In addition, the morphology of polyplexes changed from thoroidal to more elongated, worm-like forms. Some globular structures were detected in cases of a lower degree of grafting. Finally, DNA release form polyplexes when exposed to negatively charged macromolecules, poly-L-aspartic acid sodium salt, is very structure dependant. Enhanced levels of luciferase expression observed with PLL-PEG polyplexes versus either free DNA or PLL polyplexes are encouraging and warrant further optimization of the polymeric gene delivery system.