Complex homogeneous splines on the torus

We construct functions M_α which are piecewise homogeneous polynomials on the (d+1)-dimensional torus U^d+1. These functions possess complete symmetry with respect to the independent variables. The symmetry and homogeneous relations for these functions are exploited to obtain a recurrence relation and explicit representations. Furthermore, we show that , where ω=e^12x/k, 0≤j_t≤k−1, are linearly independent. By restricting M_α to U^d, we obtain the complex analogue of polynomial box splines on a (d+1)-direction mesh on U^d, which is a multivariate analogue of B-splines on the circle studied by I.J. Schoenberg^[8].     

The synthesis of 1,4-diazacycl[3,2,2]azine

The diazaanalog of “cycl[3,2,2]azine”, “1,4-diazacycl[3,2,2]azine” (1,4,7b-triazacyclopent-[cd]indene) and its 2-methyl derivative were prepared. These compounds are subject to facile acid-catalyzed hydrolysis affording substituted imidazo[1,2-a]pyridines.     

Radiation Effects on Concentrated Potassium Hexacyanoferrate(III) Solutions

The neutron irradiated potassium hexacyanoferrate(III) solutions were analyzed for Fe(CN)₆^4?, Fe(CN)₆^3?, and Prussian blue. The retention values (1.1%) were constant throughout the pH and concentration ranges studied, and did not change with the length of irradiation. The complex behavior of the radiochemical yield of the Fe(CN)₆^4? and Prussian blue was attributed to the competition of the various reactions involved in the irradiation.     

Nucleophilic substitution in 5,6,7,8-tetrafluoro- and 6,7-difluoro-1,2,3,4-tetrahydro-9-alkenyl-1,4-methanonaphthalenes and related compounds. A search for homoallylic conjugation

The second-order rate constants for the nucleophilic replacement of fluorine by isopropoxide in 5,6,7,8-tetrafluoro- and 6,7,-difluoro-1,2,3,4-tetrahydro-9-alkenyl-1,4-methanonaphthalenes and related 9-alkyl systems have been measured. A factor of 6 - 7 separates the most reactive compounds [the 9- (4′-trifluoromethylbenzylidene) derivatives] from the least reactive compounds [the syn-9-isopropyl derivatives] in both the tetrafluoro- and difluoro-series. It is concluded for these small reactivity differences.     

Strong-coupling effects in high-T c superconductors

It is conceivable that the high-T _c superconducting perovskites are conventional electronphonon superconductors. In this case one expects significant strong-coupling effects because of the unusually high ratiok _B T _c / of the order 0.1 and greater. We use a set of reasonable models for the Eliashberg function ² F() (which takes into account available information on the phonon spectra and which fit the measuredT _c 's) and calculate strong-coupling effects in the specific heatc _s (T)/T _c , the ratio ₀/k _B T _c , the critical fieldsH _c (T) andH _c2 (T) including Pauli limiting, and other measurable quantities. Strongcoupling corrections turn out to be in the range of 0 to about 100%, depending on the quantity of interest. We discuss the perspectives of using strong-coupling effects as indicators for conventional electron-phonon superconductivity in the new materials.