Limiting set of second order spectra

Let be a self-adjoint operator acting on a Hilbert space . A complex number is in the second order spectrum of relative to a finite-dimensional subspace iff the truncation to of is not invertible. This definition was first introduced in Davies, 1998, and according to the results of Levin and Shargorodsky in 2004, these sets provide a method for estimating eigenvalues free from the problems of spectral pollution. In this paper we investigate various aspects related to the issue of approximation using second order spectra. Our main result shows that under fairly mild hypothesis on the uniform limit of these sets, as increases towards , contain the isolated eigenvalues of of finite multiplicity. Therefore, unlike the majority of the standard methods, second order spectra combine nonpollution and approximation at a very high level of generality.

     

Applications of proton magnetic resonance spectroscopy to structural problems—XXXIV: Partial bond fixation in bicyclic heteroaromatic compounds

NMR spectral parameters are reported for benzofurazan, 1,2,3- and 2,1,3-benzothiadiazole, and 1-methyl- and 2-methyl-benzotriazole. The coupling constants are discussed in connection with the aromaticity of these compounds and the occurrence of partial bond fixation.     

Non-variational approximation of discrete eigenvalues of self-adjoint operators

^** Email: L.Boulton{at}ma.hw.ac.uk We establish sufficient conditions for approximation of discreteeigenvalues of self-adjoint operators in the second-order projectionmethod suggested recently in Levitin & Shargorodsky (2004,Spectral pollution and second order relative spectra for self-adjointoperators. IMA J. Numer. Anal., 24, 393–416). We findfairly explicit estimates for the eigenvalue error and studyin detail two concrete model examples. Our results show thatsecond-order projection strategies not only are universallypollution free but also achieve approximation under naturalconditions on the discretising basis.     

The tautomerism of heteroaromatic compounds with five-membered rings—VII Isoxazolin-3-ones—3-hydroxyisoxazoles

The preparation of some 3-hydroxyisoxazoles from β-keto-esters and the proof of their structure is described. These compounds are shown to exist largely in the hydroxy form.     

Ceramic films for optical applications

Recent progress is reviewed on the wet chemical synthesis of films for optical applications. Specific attention is directed to planar waveguides and to electrooptic, non-linear optical and electrochromic films, including recent results obtained in our laboratories.