On the number of triple points of the tangent developable

We study the singularities of the tangent developable of a generic curve in ³ and prove that if the curve has no torsion zero points, the number of triple points of that surface must be even.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Carbyne-fluoro complexes of rhenium. Single-pot synthesis of trans-[ReF(CCH2R)-(Ph2PCH2CH2PPh2)2][BF4]

Treatment of a THF solution of trans-[ReCl(N₂)(dppe)₂] (dppe = Ph₂PCH₂CH₂PPh₂) with a 1-alkyne HCCR (R =^tBu, CO₂Me, CO₂Et, or C₆H₄Me-4), in the presence of Tl[BF₄]/[NH₄][BF₄], under sunlight, affords the corresponding carbyne-fluoro complexes trans-[ReF(CCH₂R)(dppe)₂][BF₄] in an unprecedented single-pot synthesis. Further reaction with [BU₄N]OH leads to the vinylidenefluoro compounds trans-[ReF(=C=CHR)(dppe)₂] (R = CO₂Me, CO₂Et, or C₆H₄Me-4).     

Reactions of thiosemicarbazones derived from beta-keto amides and beta-keto esters with Zn(II) and Cd(II) acetates: influence of metal, substitution, reagent ratio and temperature on metal-induced cyclization

Zinc(II) and cadmium(II) acetates were reacted in methanol under various experimental conditions with thiosemicarbazones derived from beta-keto amides or beta-keto esters (HTSC). Some of these reactions afforded thiosemicarbazonate complexes [M(TSC)2] with IR and NMR spectra compatible with N,S-coordination, but most gave complexes [ML2], where HL is a substituted 2,5-dihydro-5-oxo-1H-pyrazole-1-carbothioamide resulting from cyclization of the HTSC. Some of these pyrazolonates and two of the HL ligands were studied by X-ray diffractometry, and their structures are discussed. Surprisingly, the reactions of zinc(II) acetate with HTSC in 1:1 mol ratio usually gave a third, previously unreported type of complex with a dideprotonated ligand, [Zn(L-H)], which was also formed when [ZnL2] and Zn(OAc)2 interacted at room temperature in 1:1 mol ratio. These L-H complexes are highly insoluble in all common solvents, which hinders their characterization but suggests that they are polymeric in nature.     

Studium des gegenseitigen Einflusses der Bindungen in Thiocyanat-Kupfer(II)-Komplexen mit Methyl- und Dimethylpyridinen mittels Infrarotspektroskopie

The infrared spectra of thiocyanato-copper(II) complexes of the type Cu(NCS)₂ L ₂ with all isomeric picolines and lutidines (L) as well as those of the type Cu(NCS)₂ L ₃ withL=3- and 4-picoline, 3,4- and 3,5-lutidine were measured. The behaviour of the (CN) stretching vibrations of the Cu(NCS)₂ L ₂ complexes led to conclusions on the different kind of coordination of the NCS groups in dependence on the ligandL. The mutually conditioned strength of the Cu–NCS and Cu–N(L) bonds in the equatorial plane, especially in the presence of ligandsL with the steric effect, was determined from the frequency shift of the Cu–NCS and Cu–N(L) stretching vibrations with regard to the vibrations for Cu(NCS)₂(pyridine)₂.

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A thieno-2H-chromene alpha-amino acid derivative: synthesis and photochromic properties

[structure: see text] A new photochromic thieno-2H-chromene alpha-amino acid derivative was prepared by C-N palladium-catalyzed cross-coupling of a bromothieno-2H-chromene with the aminated aromatic side chain of the methyl ester of a N,N-diprotected amino acid. Its good photochromic properties demonstrated by flash photolysis and continuous irradiation indicate a possible application in ophthalmic lenses. It may also be inserted into peptides to give photoinduced reversible structural changes.     

Composition dependence of the structural chemistry and magnetism of Ca2.5Sr0.5(Ga,Co)1+xMn2−xO8

Polycrystalline samples of bilayered brownmillerite-like Ca_2.5Sr_0.5GaCo_0.15Mn_1.85O₈ and Ca_2.5Sr_0.5Ga_1.2Mn_1.8O₈ have been prepared and characterised by magnetometry and neutron diffraction over a wide temperature range. The structural chemistry and magnetic properties are compared to those of Ca_2.5Sr_0.5GaMn₂O₈. Ga enrichment has a significant effect on the former but not on the latter, whereas changes in both occur when paramagnetic Co³⁺ cations enter the parent phase on the 4-coordinate sites. The coupling between the environment around the 4-coordinate cations and the transition to an antiferromagnetic ordered state that was observed in Ca_2.5Sr_0.5GaMn₂O₈ is not apparent in the cation-substituted compositions, although both show long-range antiferromagnetic order at low temperatures.     

Comprehensive study of the methyl effect on the solvolysis rates of bridgehead derivatives

The effect of a bridgehead methyl group on the hydride ion affinity in the gas phase of bicyclo[1.1.1]pent-1-yl (1+), 1-norbornyl (3+), cubyl (5+), 1-adamantyl (7+), bicyclo[2.2.2]oct-1-yl (9+),and bicyclo[3.1.1]hept-1-yl (11+) cations has been studied using density functional theory and ab initio methods. It is concluded that the methyl group always increases the stability of the substituted cations. The effect of the solvent on the stability of methyl-substituted cations in relation to the unsubstituted cations has been studied using the polarizable continuum model of the self-consistent reaction field theory. In the case of rearranging cations, the nucleophilic assistance of the solvent is determined by means of the interaction energy of the corresponding water complexes. It is concluded that the solvent causes the relative stabilization of the parent cations. As a consequence, most of the methyl-substituted bridgehead derivatives show a lower solvolysis rate than the corresponding unsubstituted compounds. A nonqualitative explanation of the methyl effect on the relative stability of bridgehead cations in both gas phase and solution is given for the first time. The ratios of solvolysis products in the case of rearranging bridgehead cations have also been computed from the relative stability of the intermediate water complexes.