Normal form for generalized Hopf bifurcation with non-semisimple 1 ∶ 1 resonance

The primary result of this research is the derivation of an explicit formula for the Poincaré-Birkhoff normal form of the generalized Hopf bifurcation with non-semisimple 1:1 resonance. The classical nonuniqueness of the normal form is resolved by the choice of complementary space which yields a unique equivariant normal form. The 4 leading complex constants in the normal form are calculated in terms of the original coefficients of both the quadratic and cubic nonlinearities by two different algorithms. In addition, the universal unfolding of the degenerate linear operator is explicitly determined. The dominant normal forms are then obtained by rescaling the variables. Finally, the methods of averaging and normal forms are compared. It is shown that the dominant terms of the equivariant normal form are, indeed, the same as those of the averaged equations with a particular choice for the constant of integration.Partially supported by NSF through grant MSS 90-57437, AFOSR through grant 91-0041 and NSERC of Canada.     

Synthesis of (diarylmethyl)amines using Ni-catalyzed arylation of C(sp3)–H bonds

The first nickel catalyzed deprotonative cross coupling between C(sp³)–H bonds and aryl chlorides is reported, allowing the challenging arylation of benzylimines in the absence of directing group or stoichiometric metal activation. This methodology represents a convenient access to the (diarylmethyl)amine moiety, which is widespread in pharmaceutically relevant compounds.     

Competitive Gold‐Promoted Meyer–Schuster and oxy‐Cope Rearrangements of 3‐Acyloxy‐1,5‐enynes: Selective Catalysis for the Synthesis of (+)‐(S)‐γ‐Ionone and (−)‐(2S,6 R)‐cis‐γ‐Irone

We report a simple, highly stereoselective synthesis of (+)‐(S)‐γ‐ionone and (‐)‐(2S,6R)‐cis‐γ‐irone, two characteristic and precious odorants; the latter compound is a constituent of the essential oil obtained from iris rhizomes. Of general interest in this approach are the photoisomerization of an endo trisubstituted cyclohexene double bond to an exo vinyl group and the installation of the enone side chain through a [(NHC)Au^I]‐catalyzed Meyer–Schuster‐like rearrangement. This required a careful investigation of the mechanism of the gold‐catalyzed reaction and a judicious selection of reaction conditions. In fact, it was found that the Meyer–Schuster reaction may compete with the oxy‐Cope rearrangement. Gold‐based catalytic systems can promote either reaction selectively. In the present system, the mononuclear gold complex [Au(IPr)Cl], in combination with the silver salt AgSbF₆ in 100:1 butan‐2‐one/H₂O, proved to efficiently promote the Meyer–Schuster rearrangement of propargylic benzoates, whereas the digold catalyst [{Au(IPr)}₂(μ‐OH)][BF₄] in anhydrous dichloromethane selectively promoted the oxy‐Cope rearrangement of propargylic alcohols.     

Well-defined [Rh(NHC)(OH)] complexes enabling the conjugate addition of arylboronic acids to α,β-unsaturated ketones

The synthesis and catalytic activity of three well-defined monomeric rhodium(I) hydroxide complexes bearing N-heterocyclic carbene (NHC) ligands are reported. [Rh(cod)(ICy)(OH)] promoted the 1,4-addition of arylboronic acids to cyclic enones, with TONs and TOFs of 100,000 and 6,600 h(-1), respectively, at 0.001 mol% catalyst loadings. Mechanistic studies permitted the isolation of a phenylrhodium intermediate.     

Catalytic deuteration of silanes mediated by N-heterocyclic carbene-Ir(III) complexes

The catalytic activity of a series of coordinatively unsaturated NHC-M(III) (M = Rh, Ir; NHC = N-heterocyclic carbene) complexes was tested in the deuteration of secondary and tertiary silanes. Among these, [IrCl(I(t)Bu')(2)] provides the highest conversions to the deuterated species. Mechanistic studies highlight the reversible nature of the ortho-metalation reaction.     

Yang's Gravitational Theory

Yang's pure space equations generalize Einstein's gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors.     

High-spin study of Xe

High-spin states have been populated in ₅₄¹¹⁹Xe via the ⁹⁶Mo(²⁷Al,p3n) reaction at 133 MeV, using the γ-ray spectrometer to record triple γ-ray coincidences. The known level scheme has been significantly extended and several band crossings identified. In particular, the νh_11/2 yrast band has been extended to I^π = (83/2⁻) and shows features which are consistent with those of smooth band termination at high spin. Theoretical results for ¹¹⁹Xe at high spin are discussed within the framework of cranked Nilsson-Strutinsky calculations, together with results for ¹¹⁷Xe.