Embeddings and hyperplanes of the Lie incidence geometry of type E7,1

This paper continues a program to show that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily absolutely universal embedding, by addingE _7,1 to the list. It follows from [5, 12] that any projective embedding of this point line geometry is a homomorphic image of the one afforded by the 56-dimensional module for the groupE ₇(K).This work was supported by a grant from the National Science Foundation.     

A rigid graph for every set

A graph G is called rigid if the identical mapping V(G)→V(G) is the only homomorphism G→G. In this note we give a simple construction of a rigid oriented graph on every set. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 108–110, 2002     

Aluminum(III) complexes containing O,O chelating ligand

The stoichiometric reactions of trimethylaluminum with 2,6‐(MeOCH₂)₂C₆H₃OH (LH) revealed compounds L₃Al ( 1 ) and L₂AlMe ( 2 ). On the other hand reaction of 1 equiv. of LH with trimethylaluminum did not lead to the formation of complex LAlMe₂ ( 3 ), rather 2 together with Me₃Al were observed as a result of a disproportionation of 3 . Compounds 1 and 2 were characterized by elemental analysis, ¹H and ¹³C NMR spectroscopy and in the case of 1 by X‐ray diffraction. Derivative 2 underwent transmetalation with Ph₃SnOH, giving LSnPh₃ ( 4 ) as the result of a migration of ligand L from the aluminum to the tin atom. The identity of 4 was established by elemental analysis, ¹H, ¹³C and ¹¹⁹Sn NMR spectroscopy and ¹H, ¹¹⁹Sn HMBC experiments. The system 2 and B(C₆F₅)₃ in a 1:1 molar ratio was shown to be active in the polymerization of propylene oxide and ε‐caprolactone. Copyright © 2007 John Wiley & Sons, Ltd.     

2-Oxocephems. Synthesis and Properties of 7-Acylamino-2-oxocephem-4-carboxylic Acids

A synthesis of (6 R, 7 R)-7-acylamino-2-oxocephem-4-carboxylic acids from penicillins is reported based on the formation of the 2-oxocephem system in an intramolecular Wittig condensation. An interesting steric effect of the acylamino grouping on the course of this key step was used to advantage. Some physical and biological properties of this new type of Δ³-cephem are described.     

Chemical potential for harmonically interacting particles in a harmonic potential

The chemical potential μ, as it appears in density functional theory, is examined extensively for harmonically interacting spin-½ fermions in three dimensions. For this system the energy and chemical potential are discontinuous functions of the particle number if the most straightforward equation is used to define the energy for a noninteger particle number.     

Rotation and inversion barriers in 2-lithio-2-phenyl-1,3-dithianes

The barriers to phenyl rotation in 2-lithio-2-phenyl-cis-4,6-dimethyl-, 2-lithio-2-phenyl-4,4,6-trimethyl- and 2-lithio-2-phenyl-trans-4,6-dimethyl-1,3-dithiane are compared in tetrahydrofuran (THF) and hexamethylphosphortriamide (HMPA). In the first two cases, the barriers in THF are lower than those in HMPA, presumably because the lithio compound exists as a tight ion pair in THF but as a solvent-separated ion pair (with more delocalization of charge into the phenyl ring) in HMPA. However, in the trans-4,6-dimethyl compound the barriers are the same in the two solvents and nearly equal to the barriers for ring reversal. It is concluded that in this compound the rate-determining step for phenyl rotation may actually be ring reversal, at least in solvent HMPA.