An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87)for analysing the time discretization error and assuming moreregularity on the initial data, we improve on the error boundderived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16,257-287) for a fully practical piecewise linear finite elementapproximation with a backward Euler time discretization of amodel for phase separation of a multi-component alloy.     

Poisson structures due to Lie algebra representations

Using a unitary solution of the classical Yang-Baxter equation on a Lie algebraG we describe a particular way of constructing homogeneous quadratic Poisson structures on the dual of aG-moduleV and study some local features of the symplectic foliation due to the involutive distribution of the Hamiltonian vector fields. We also give some examples where the symplectic leaves are explicitly calculated.     

Eigenvalues of the form valued Laplacian for Riemannian submersions

Let be a Riemannian submersion of closed manifolds. Let be an eigen -form of the Laplacian on with eigenvalue which pulls back to an eigen -form of the Laplacian on with eigenvalue . We are interested in when the eigenvalue can change. We show that , so the eigenvalue can only increase; and we give some examples where , so the eigenvalue changes. If the horizontal distribution is integrable and if is simply connected, then , so the eigenvalue does not change.

     

A synthetic approach to the phorboxazoles – A strategy for the synthesis of the C1–C19 polyketide fragment

A synthetic approach to the C1–C19 polyketide fragment of the phorboxazoles is disclosed here. While an initial two-directional approach was efficient, it did not proceed in a high enough yield to justify moving forward. A subsequent successful strategy for the generation of the C11–C15 pyrans of both of the phorboxazoles was achieved, and the installation of the C9 stereocenter was able to be demonstrated. Furthermore, an efficient route for the preparation of the C1–C8 fragment with suitable functionality to allow for elaboration into the complete C1–C19 fragment, with the capricious C2–C3 Z-geometry installed, was also achieved.