A Prüfer angle approach to singular Sturm–Liouville problems with Molčanov potentials

Singular Sturm–Liouville problems for -y^″+qy=λy$-y^{″}+\mathit{qy}=\lambda y$ on (0,∞)$(0,\infty )$ are studied for potentials q which are bounded below and satisfy Mol?anov's necessary and sufficient condition for discrete spectrum. A Prüfer angle approach is given for eigenvalue location and eigenfunction oscillation, paralleling that for the regular case. In particular, the eigenvalues are characterized by a “right-hand boundary condition” even though q is of limit point type.     

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.     

Ferroelectric liquid crystals induced by dopants with axially chiral 2,2'-spirobiindan-1,1'-dione cores

The axially chiral dopants (R)-5,5'-, 5,6'-, and 6,6'-diheptyloxy-2,2'-spirobiindan-1,1'-dione ((R)-2, -3, and -4) were synthesized in optically pure form, and their absolute configurations were assigned by the exciton chirality method using circular dichroism spectroscopy. These new compounds were doped in four achiral liquid crystal hosts to give chiral smectic C* (SmC*) phases with spontaneous polarizations (Ps) that vary with the core structure of the host. The spontaneous polarization induced by the 5,5'-dialkoxy derivative (R)-2 is uniformly positive, whereas that induced by the 6,6'-dialkoxy derivative (R)-4 is uniformly negative and shows a different trend in host dependence. Polarization power (delta(p)) values range from +21 nC/cm2 for (R)-2 in 2',3'-difluoro-4-heptyl-4' '-nonyl-p-terphenyl (DFT) to -1037 nC/cm2 for (R)-4 in 4-(4'-heptyl[1,1'-biphen]-4-yl)-1-hexylcyclohexanecarbonitrile (NCB76). The unsymmetrical dopant (R)-3 behaves like a hybrid of the two symmetrical isomers, with lower absolute values of delta(p), on average, and varying signs of Ps. 2H NMR spectra of the doped mixtures using racemic mixtures of 2-4 with -OCD2C6H13 side-chains, in combination with phase diagrams, show that relatively minor changes in the dopant structure, that is, moving the alkoxy side-chains from the 5,5' to the 6,6' positions of the spirobiindandione core, have profound effects on dopant-host compatibility, and on the propensity of the dopant to exert chiral perturbations in the host environment. The variations in sign and magnitude of delta(p) as a function of alkoxy group positions are rationalized based on an analysis of zigzag conformations that conform to the binding site of the SmC host according to the Boulder model.     

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.