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     

Acyl- und Alkylidenphosphane. XXVIII. Synthese und Struktur von 1,3-Dibenzyl- und 1,3-Diethyl-2,4-bis(phenylimino)-1,3-diphosphetan

Acyl- and Alkylidenephosphines. XXVIII. Synthesis and Structure of 1,3-Dibenzyl- and 1,3-Diethyl-2,4-bis(phenylimino)-1,3-diphosphetane Catalyzed by small amounts of solid sodium hydroxide, the adducts 1a and 1b formed from benzyl- or ethylbis(trimethylsilyl)phosphine and phenylisocyanate, react at +20°C slowly to give hexamethldisiloxane and oligomeric [(phenylimino)methylidene]phosphines. In different solvents the benzyl compound was found to exist only as a mixture of [N,N′-(E)/(Z)]-isomeric 2,4-bis-(phenylimino)-1,3-diphosphetanes 2a with their alkyl groups at the phosphorus atoms in trans position, whereas in case of the ethyl derivative 2b a second pair of [N,N′-(E)/(Z)]-isomeric dimers with their substituents in cis position and two trimeric forms ( 3b and 4b ) could be detected in cyclopentane. [N,N′-(E)]-1r,3t-dibenzyl- ( 2a ) and [N,N′-(E)]-1r,3t-diethyl-2,4-bis(phenylimino)-1,3-diphosphetane 2b isolated from 1,2-dimethoxyethane or cyclopentane, crystallize in the monoclinic space group P2₁/c or P2₁/n, resp., with following dimensions of the unit cell determined at temperatures of measurement of +20 ± 3°C/?130 ± 3°C: a = 2145.4(1)/569.3(1); b = 568.1(2)/719.1(2); c = 1960.2(2)/2042.6(4) pm; β 99.43(1)°/95.03(2)°; Z = (2+2) and 2, resp. X-ray structure determinations (R_w = 0.034/0.041) show both molecules to be centrosymmetric. Characteristic rounded bond lengths (pm) and angles (°) are: endocyclic P? C 185/184; C? P? C 82/81; P? C? P 98/99; exocyclic P? C 186/184; C?N l27/127; C?N? C 121/11.     

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.     

Synthesis and molecular structures of dialkylselenonium methylide complexes of indium tribromide

The reaction of bromomethyl-dibromo-indium(III), Br₂InCH₂Br with dialkylselenides, R¹SeR² (R¹ = CH₃, R² = CH₂C₆H₅; R¹ = C₂H₅, R² = CH₂C₆H₅; R¹ = R² = CH₂C₆H₅) afforded the corresponding dialkylselenonium methylide complexes of indium tribromide, Br₃InCH₂SeR¹R², which were fully characterized by NMR spectroscopy and single crystal X-ray diffraction studies.     

Acyl- und Alkylidenphosphane. XXIII [1]. Synthese und Struktur der [Bis(trimethylsilylsulfano)methyliden]phosphane

Acyl- and Alkylidenephosphines. XXIII. Synthesis and Structure of [Bis(trimethylsilylsulfano)methylidene]phosphines Analogous to the phenyl derivative 1a [2] tert-butyl- 1b , mesityl- 1c and methylbis-(trimethylsilyl)phosphine 1 d react with carbon disulfide to give the corresponding [bis(trimethylsilylsulfano)methylidene]phosphines 4 . Only in case of the mesitylphosphine 1 c the intermediate compounds 2 and 3 could be detected by n.m.r. spectroscopic methods; thermally unstable [bis(trimethylsilylsulfano)methylidene]methylphosphine 4 d dimerizes rapidly [1]. [Bis(trimethylsilylsulfano)methylidene]phenylphosphine 4 a crystallizes in the monoclinic centrosymmetric space group P2₁/c with following dimensions of the unit cell determined at ?95 ± 3°C: a = 1386.4(8); b = 1036.0(7); c = 1281.7(8) pm; ß = 101.23(4)°; Z = 4. An X-ray structure determination (R = 0.032) proves the constitution of this compound as already derived from its nmr spectra. Characteristic bond lengths and angles are: P?C 170; P? C(phenyl) 183; C? S 176; S? Si 219 pm; C? P?C 107; P?C? S 124 and 120; S? C? S 116 and C? S? Si 111°.     

The ferromagnetic and paramagnetic properties of heusler alloys (Ni1−xCox)2MnSn

Mixed Heusler alloys with composition (Ni_1?xCo_x)₂MnSn (x = 0–1). were prepared. They were single phase with L2₁ structure. Magnetic measurements were undertaken in the ferromagnetic and paramagnetic regions. Magnetic parameters, such as Curie temperature, paramagnetic Curie temperature, and spontaneous moment μ₀₀ depend linearly on the composition. The linear variation of μ₀₀ is phenomenologically interpreted by a linear variation of the magnetic moment of manganese between 4.0 μ_B for Ni₂MnSn to 3.6 μ_B for Co₂MnSn, cobalt having a constant moment of 0.75 μ_B over the whole concentration range.