Complex powers of classical SG-pseudodifferential operators

Abstract Under a suitable ellipticity condition, we show that classical SG-pseudodifferential operators of nonnegative order possess complex powers. We show that the powers are again classical and derive an explicit formula for all homogeneous components. Keywords: Complex power, Weighted symbols, Noncompact manifolds     

Bildung und Kristallstruktur eines dimeren sauerstoffverbrückten Titanaminobisphenoxides

Formation and Crystal Structure of an Oxygen Bridged Titanium Amino Bisphenoxide Di‐(μ‐oxo)‐titan‐bis[aminobisphenoxide] ( 3 ) was obtained by reaction of (i‐Prop)₂TiL* ( 2 ) {L* = O‐4, 6‐(t‐Bu)₂C₆H₂‐2‐CH₂‐[1, 4‐N₂C₅H₁₀]‐2'‐CH₂‐4', 6'‐(t‐Bu)₂C₆H₂O] with water in a molar ratio 1:1 in diethylether. Both i‐Propyl moieties are substituted yielding the dimeric oxygen bridged amino bisphenoxide complex. The six coordinate Ti atoms are a result of Ti—N and Ti—O interactions of the corresponding atoms of the amino bisphenoxide ligand and the bridging O atoms. The central planar Ti₂O₂ ring may be considered as the general structural feature of the title compound: Space group P1¯, Z = 1, lattice dimensions at —60°C: a = 11.6899(4), b = 11.7873(4), c = 12.6462(4) Å, α = 98.070(1), β = 99.660(1), γ = 95.343(1)°, R₁ = 0.0469, wR₂ = 0.1049, GooF = 0.939.     

Uranyl ion-selective optode based on neutral ionophores

A pulsed molecular beam Fourier transform microwave spectrometer especially designed for the quantitative analysis of polar pollutants in air is presented. This very fast method has a sensitivity in the ppm range and can be applied to a wide variety of substances. Details of the apparatus and the results of some performance tests are given.     

EVIDENCE FOR THE EXISTENCE OF MEMBRANE-ASSOCIATED PHYTOCHROME IN THE CELL

Abstract Comparative fluorescence and photochemical studies of phytochrome in etiolated seedlings of maize and in soluble and membrane-containing fractions isolated from them were camed out. The membrane fractions prepared in the absence of Mg²⁺ from etiolated coleoptiles contained 13% of total photoreversible phytochrome, which was readily solubilized by mild detergents. Its molecular size was indistinguishable from soluble phytochrome and equal to nondegraded maize phytochrome. Low-temperature fluorescence studies with intact tissue found that the position of the emission maximum at 85 K (λ_max) and the extent of the phototransformation of the red-absorbing form (Pr) into the first stable photoproduct, lumi-R, at 85 K (γ₁), varied in different parts of etiolated seedlings: λ_max and γ₁ reached their maximum values in the tips of coleoptiles and roots, 686 nm and 0.30–0.40, whereas the lowest values, 682 nm and ca 0.05, were observed in the root base. These parameters correlated well with those obtained for the pigment in the soluble and membrane-containing fractions: 684 and 680 nm, and 0.33 and 0.06, respectively. The extent of the Pr phototransformation into the far red-absorbing form (Pfr) (γ₂) did not differ much: values of 0.80–0.85 and 0.70–0.75 correlated with the high and low values of γ₁. These variations of the parameters were interpreted in agreement with our previous observations in terms of two phytochrome A species whose relative concentrations vary depending on the experimental conditions—the longer wavelength bulk light-labile species with high γ₁ (Pr″), and the shorter wavelength minor light-stable species with low γ₁ (Pr″). Close similarity between Pr’and the soluble phytochrome and between Pr″ and the membrane-bound phytochrome points to the possible origin of the native Pr’and PrPrime; species, thus providing evidence for the existence of membrane-bound pigment in the cell.     

Invariance of the cone algebra without asymptotics

Let B be a manifold with conical singularities, and denote by the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.B.-W. Schulze has developed a framework for a pseudodifferential calculus on B by defining various classes of distribution spaces and operator algebras, working in fixed coordinates on the manifold . I am showing here that the Mellin Sobolev spaces without asymptotics, the cone algebra without asymptotics, and its ideal of smoothing operators are independent of the choice of coordinates and therefore may be considered intrinsic objects for manifolds with conical singularities.     

On the symbol homomorphism of a certain Frechet algebra of singular integral operators

We prove the surjectivity of the symbol map of the Frechet algebra obtained by completing an algebra of convolution and multiplication operators in the topology generated by all L²-Sobolev norms. The proof is based on an ⁿ of Egorov's theorem valid for non-homogeneous principal symbols, discussed in [5], [6]. We use the hyperbolic equation u/t=i|D|u, 0<<1, which has its characteristic flow constant at infinity, so that no differentiability of the symbol is required there.