Homothetic and conformal symmetries of solutions to Einstein's equations

We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.     

Migratory-insertion reactions involving the thiocarbonyl ligand. dihapto-thioacyl complexes from the rearrangement of arylthiocarbonyl complexes of osmium(II). Structure of Os(η2-CSR)(η1-O2CCF3)(CO)(PPh3)2

Reaction of HgR₂ with OsHCl(CS)(PPh₃)₃ yields red, five-coordinate, OsRCl-(CS)(PPh₃)₂ (R = p-tolyl). From this have been derived the compounds OsRX(CS)(PPh₃)₂ with X = Br, I, S₂CNEt₂, O₂CMe, O₂CCF₃. These compounds add an additional ligand, MeCN, CO or CNR to form colourless, six coordinate arylthiocarbonyl complexes, which undergo migratory-insertion reactions to form red, dihapto-thioacyl complexes. The crystal structure of a representative example, Os(η²-CSR)(η¹-O₂CCF₃)(CO)PPh₃)₂ has been determined. The red equant crystals are orthorhombic, space group P2₁2₁2₁, a 11.584(1), b 19.184(2), c 18.90(1) Å, V 4199 ų, Z  4. The structure was solved by conventional heavy-atom methods and refined by full-matrix least-squares employing anisotropic thermal parameters for all non-hydrogen atoms except the carbon atoms of the triphenylphosphines. The final R factor is 0.057 for 2868 observed reflections.The coordination geometry in the monomeric complex is that of an octahedron distorted by the constraints of the ligands. The triphenyl phosphine ligands are mutually trans; the equatorial plane contains carbonyl, monohapto-trifluoroacetate, and dihapto-thioacyl ligands. Bond distances and angles are OsP 2.405, 2.407(4) Å; POsP 173.9(1)°; OsCO 1.83(2) Å; Os-O (trifluoroacetate) 2.206(11) Å; OsC (thioacyl) 1.91(2); OsS 2.513(6); CS 1.72 Å. The CS bond length implies a reduction in bond order from 2.0 to approx. 1.5 upon coordination to the metal.The η²-thioacyl ligand in Os(η²-CSR)Cl(CNR)(PPh³)₂ is methylated with methyl triflate and further reaction with LiCl produces the thiocarbene complex OsCl₂(C[SMe]R)(CNR)(PPh₃)₂.     

Structure of a fluorinated azoxy compound: fluoro(trifluoromethyl)-diazene-2-oxide,CF3N(O)NF

A gas-phase electron diffraction study of the azoxy compound which was synthesized by the reaction of CF3NO with N2F4 in a Pyrex glass vessel results in a trans CF3N(O)NF structure (F trans to CF3), although quantum chemical calculations (MP2 and B3LYP) predict a greater stability of the cis CF3NN(O)F isomer by about 12 kcal/mol. The CF3 group eclipses the N=N double bond. The following skeletal geometric parameters (r(a) values with 3sigma uncertainties) were obtained: N=N 1.287(15) A; N=O 1.231(6) A; N-F 1.380(6) A; N-C 1.498(6) A; N=N=O 131.2(13) degrees; N=N-F 103.5(13) degrees; N=N-C 114.0(12) degrees. The bond lengths in CF3N(O)NF are compared to those in azo, nitryl, and nitrosyl compounds with fluorine and/or CF3 substituents.     

Picolinamides as Effective Ligands for Copper‐Catalysed Aryl Ether Formation: Structure–Activity Relationships,Substrate Scope and Mechanistic Investigations

The use of picolinic acid amide derivatives as an effective family of bidentate ligands for copper‐catalysed aryl ether synthesis is reported. A fluorine‐substituted ligand gave good results in the synthesis of a wide range of aryl ethers. Even bulky phenols, known to be very challenging substrates, were shown to react with aryl iodides with excellent yields using these ligands. At the end of the reaction, the first examples of end‐of‐life Cu species were isolated and identified as Cu^II complexes with several of the anionic ligands tested. A preliminary mechanistic investigation is reported that suggests that the substituents on the ligands might have a crucial role in determining the redox properties of the metal centre and, consequently, its efficacy in the coupling process. An understanding of these effects is important for the development of new efficient and tunable ligands for copper‐based chemistry.     

Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.Research partially supported by ARO Contract DAAG-29-79-C-0086 and by NSF Grants ENG 78-02891 and MCS-78-06718