Results and Estimates on Multiple Solutions of Lidstone Boundary Value Problems

We consider the following boundary value problem (-1)ⁿy(²ⁿ)=F(t,y), n≥ 1, t ∈ (0,1), y⁽²ⁱ⁾(0)=y⁽²ⁱ⁾(1)=0, 0≧i≧n-1. Criteria are developed for the existence of two and three positive solutions of the boundary value problem. In addition, for special cases we establish upper and lower bounds for these positive solutions. Several examples are also included to dwell upon the importance of the results obtained.     

Pseudostress–velocity formulation for incompressible Navier–Stokes equations

This paper presents a numerical algorithm using the pseudostress–velocity formulation to solve incompressible Newtonian flows. The pseudostress–velocity formulation is a variation of the stress–velocity formulation, which does not require symmetric tensor spaces in the finite element discretization. Hence its discretization is greatly simplified. The discrete system is further decoupled into an H ( div ) problem for the pseudostress and a post‐process resolving the velocity. This can be done conveniently by using the penalty method for steady‐state flows or by using the time discretization for nonsteady‐state flows. We apply this formulation to the 2D lid‐driven cavity problem and study its grid convergence rate. Also, computational results of the time‐dependent‐driven cavity problem and the flow past rectangular problem are reported. Copyright © 2009 John Wiley & Sons, Ltd.     

Convenient one-pot synthesis of trans-1,2-diaryloxiranes from the direct coupling of benzyl halides

trans-1,2-Diaryloxiranes were conveniently prepared in an one-pot reaction by the direct coupling of benzyl halides in the presence of silver oxide and DMSO under mild conditions.     

Hydroxyl Protection by Levulinylation

The active hydrogen of the hydroxy group is often masked by ester formation. To avoid difficulties sometimes arise during regeneration of the alcohol, a number of functionalized carboxylic acids may be employed for esterification. The derived esters should permit mild and selective cleavage methods other than saponification to be used.