Let $\go{g}$ be a finite-dimensional semi-simple Lie algebra, $\go{h}$ a Cartan subalgebra of $\go{g}$, and $W$ its Weyl group. The group $W$ acts diagonally on $V:=\go{h}\oplus\go{h}^*$, as well as on $\mathbb{C}[V]$. The purpose of this article is to study the Poisson homology of the algebra of invariants $\mathbb{C}[V]^W$ endowed with the standard symplectic bracket. To begin with, we give g...

In this paper, we present a numerical scheme based on collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is generalized version of the arising from form biomolecular modeling case. Applying radial basis functions (RBFs) allows us deal with difficulties complexity domain. To indicate accuracy RBF method, results for two-dimensional models, also stu...

In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introdu...

In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introdu...

In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introdu...

In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introdu...

In the remainder, we call such a variety a convex symplectic variety. A convex symplectic variety has been studied in [K-V], [Ka 1] and [G-K]. One of main difficulties we meet is the fact that tangent objects TX and T 1 Y are not finite dimensional, since Y may possibly have non-isolated singularities; hence the usual deformation theory does not work well. Instead, in [K-V], [G-K], they introdu...

This paper is the first of two papers on the adaptive multilevel finite element treatment of the nonlinear Poisson-Boltzmann equation (PBE), a nonlinear elliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, ...

Efficiency and accuracy are two major concerns in numerical solutions of the Poisson-Boltzmann equation for applications in chemistry and biophysics. Recent developments in boundary element methods, interface methods, adaptive methods, finite element methods, and other approaches for the Poisson-Boltzmann equation as well as related mesh generation techniques are reviewed. We also discussed the...