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.     

On sparse spanners of weighted graphs

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v V, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.The work of G. Das and D. Joseph was supported by NSF PYI Grant DCR-8402375. The work of D. Dobkin was supported by NSF Grant CCR-8700917. The work of J. Soares was supported by CNPq proc 203039/87.4 (Brazil) and NSF Grant CCR-9014562. This research was accomplished while G. Das was a student at the University of Wisconsin-Madison. A preliminary version was presented at the Second Scandinavian Workshop on Algorithm Theory, Bergen, Norway, 1990, under the title Generating Sparse Spanners for Weighted Graphs, and proceedings appear in the series Lecture Notes in Computer Science, Springer-Verlag. The preliminary version also appears as Princeton University Technical Report CS-TR-261-90, and as University of Wisconsin-Madison Computer Sciences Technical Report 882.     

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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