Numerical ranges of weighted composition operators

The operator that takes the function f   to ψf°φ$\psi f°\phi$ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H²$H^2$, the Hardy–Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.     

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.     

On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem

Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L ²-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.     

Application of a process friendly morpholine synthesis to (S,S)-Reboxetine

We report our results on the construction of a morpholine ring system from the corresponding epoxide and amino alcohol. From this study, we were able to convert a previous four-step synthesis into a more efficient two-step process.     

Dynamic non-linear optical processes in semiconductors

In this article active non-linear optical effects in semiconductors are reviewed. These processes arise because of the dynamics of excited populations of charge carriers temporarily created in otherwise empty states when a beam of laser radiation is incident on the material. A large number of different effects have been observed in different semiconductor materials recently and the non-linearities cover a wide range of magnitudes both in time-scale and size of non-linearity. The theory and physical concepts relevant to these processes are described and experimental observations using both high power pulsed and low power c.w. lasers are reviewed. Applications in optical bistability, phase conjugation, optical gating and optoelectronic gating are discussed.