An analytical proof of Hardy-like inequalities related to the Dirac operator

We prove some sharp Hardy-type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.     

Long-Range proton-proton coupling constants. I. Propanic coupling involving a methyl group 4JMeH

An equation is formulated on the basis of theoretical INDO/FPT calculations which describes the angular dependence of the propanic long-range coupling constant ⁴J_MeH in substituted HCCCH₃ fragments. This equation is a truncated Fourier series in the torsion angle ?, HCCMe, which takes into account the dependence of the Fourier coefficients on the bond angle θ, CCMe. The substituent effects are assumed to be additive. Some parameters in the equation may be obtained from the ⁴J_MeH couplings in propane and neopentane derivatives. The calculated effect upon ⁴J_MeH of changes in the bond angle θ is significant and it seems to be in part the cause of some effects which have been attributed to conformational dependence.     

Synthesis and Characterization of Heterodinuclear IrCo, RuCo, IrNi, and RuNi Complexes Containing Pyrazolate and Pyrazolylborate Ligands

Treatment of the metallo ligands [ML(pz)(2)(Hpz)] (pz = pyrazolate; L = C(5)Me(5), M = Ir (1); L = mesitylene, M = Ru (3)) with [M'Cl{HB(3-i-Pr-4-Br-pz)(3)}] (M' = Co (4), Ni (5)) yields heterodinuclear complexes of formula [LM(&mgr;-pz)(2)(&mgr;-Cl)M'{HB(3-i-Pr-4-Br-pz)(3)}] (L = C(5)Me(5); M = Ir; M' = Co (6), Ni (7). L = mesitylene; M = Ru; M' = Co (8)). The related complex [Ru(eta(6)-p-cymene)(pz)(2)(Hpz)] (2) reacts with equimolar amounts of 4 or 5 to give mixtures of the corresponding bis(&mgr;-pyrazolato) &mgr;-chloro complexes [(eta(6)-p-cymene)Ru(&mgr;-pz)(2)(&mgr;-Cl)M'{HB(3-i-Pr-4-Br-pz)(3)}] (M' = Co (9), Ni (10)) and the triply pyrazolato-bridged complexes [(eta(6)-p-cymene)Ru(&mgr;-pz)(3)M'{HB(3-i-Pr-4-Br-pz)(3)}] (M' = Co (11), Ni (12)). Complex 1 reacts with 5 in the presence of KOH to give the IrNi complex [(eta(5)-C(5)Me(5))Ir(&mgr;-pz)(3)Ni{HB(3-i-Pr-4-Br-pz)(3)}] (13) whereas its reaction with 4 and KOH rendered the bis(&mgr;-pyrazolato) &mgr;-hydroxo complex [(eta(5)-C(5)Me(5))Ir(&mgr;-pz)(2)(&mgr;-OH)Co{HB(3-i-Pr-4-Br-pz)(3)}] (14). The molecular structure of the heterobridged IrCo complex (6) and those of the homobridged RuNi (12) and IrNi (13) complexes have been determined by X-ray analyses. Compound 6 crystallizes in the monoclinic space group P2(1)/n, with a = 10.146(5) ?, b = 18.435(4) ?, c = 22.187(13) ?, beta = 97.28(4) degrees, and Z = 4. Complex 12 is monoclinic, space group P2(1), with a = 10.1169(7) ?, b = 21.692(2) ?, c = 11.419(1) ?, beta = 112.179(7) degrees, and Z = 2. Compound 13 crystallizes in the monoclinic space group Cc, with a = 13.695(2) ?, b = 27.929(6) ?, c = 13.329(2) ?, beta = 94.11(4) degrees, and Z = 4. All the neutral complexes 6, 12, and 13 consist of linear M.M'.B backbones with two (6) or three (12, 13) pyrazolate ligands bridging the dimetallic M.M' units and three substituted 3-i-Pr-4-Br-pz groups joining M' to the boron atoms. The presence in the proximity of the first-row metal M' of the three space-demanding isopropyl substituents of the pyrazolate groups induces a significant trigonal distortion of the octahedral symmetry, yielding clearly different M'-N bond distances on both sides of the ideal octahedral coordination sphere of these metals.     

31P NMR spectroscopy as a powerful tool for the determination of enantiomeric excess and absolute configurations of alpha-amino acids

An easy method for the determination of the enantiomeric excess (ee) of mixtures of alpha-amino acids, and also for the elucidation of the absolute configuration of each component of the mixture, is reported. The method is based on the formation of diastereoisomers by reaction of the enantiomerically pure acetylacetonate derivative [Pd(acac-O,O')(P(2)-dach)]ClO(4) (4) [P(2)-dach = (1R,2R)-C(6)H(10)(NHPPh(2))(2)] with d,l-mixtures of alpha-amino acids AaH (Pd:AaH = 1:1 molar ratio, refluxing MeOH). The reaction occurs with protonation of the acac ligand and N,O-coordination of the amino acidate group, giving the corresponding [Pd(Aa-N,O)(P(2)-dach)]ClO(4) complexes l-5 and d-6. The composition of these mixtures of amino acidate complexes was analyzed by integration of the corresponding peaks (four doublets, two for each diastereomer) in their (31)P((1)H) NMR spectra. A series of 14 alpha-amino acids was studied (a, alanine; b, 2-aminobutyric acid; c, valine; d, phenylalanine; e, proline; f, leucine; g, isoleucine; h, norleucine; i, serine; j, threonine; k, methionine; l, aspartic acid; m, glutamine; n, cysteine), and excellent agreement between the expected values of ee and those obtained from integration of the (31)P((1)H) NMR spectra was obtained. Moreover, the position of the signals of each isomer is diagnostic, in such a way that the outer doublets are always due to the l-derivatives 5a-l, while the inner ones are due to the d-derivatives 6a-l, allowing the assignation of absolute configurations to each isomer in the mixture.     

Geometry and the Jones projection of a state

LetA be a von Neumann algebra and a faithful normal state. ThenO = { ºAd(g ¹) :g G _A }andU = { ºAd(u ^*) :u U _A are homogeneous reductive spaces. IfA is aC ^* algebra,e the Jones projection of the faithful state viewed as a conditional expectation, then we prove that the similarity orbit ofe by invertible elements ofA can be imbedded inAA in such a way thate is carried to 1 1 and the orbit ofe to a homogeneous reductive space and an analytic submanifold ofAA.     

Synthesis of discodermolide intermediates from engineered polyketides

Key intermediates used in Smith’s total synthesis of discodermolide were synthesized from an engineered polyketide made via precursor feeding to genetically modified polyketide producing bacteria.     

Enantioselective Synthesis of 3- Azabicyclo[4.3.0]nonane Alkaloids

The stereospecific synthesis of the monoterpene alkaloids (?)-α-skytanthine ((?)- 2 ), (?)-N -demethyle-δ-sky-tanthine((?)- 7 ), and (+)-epidihydrotecomanine (+)- 4 was achieved from a common intermediate 22 , which in turn was obtained from (1R,4S,1′S)-2-(1′-phenylethyl)-2-azabicyclo[2.2.1]hept-5-ene (10) ,via a ketene aza-Claisen rearrangement. The piperidine derivative (+)- 31 , formally the aza-analogue of (+)-isoiridomyrmecin, was also obtained from the same intermediate 22 .     

Absolute Configuration of 3-Substituted 1-Azabicyclo[2.2.1]heptanes

The l-azabicyclo[2.2.1]heptan-3-exo-ol ( 2 ) was resolved by fractional crystallisation of its hydrogen tartrate salts. The enantiomers (+)- and (?)- 2 were oxidised to the ketones (?)- 4 and (+)- 4 , respectively (Scheme). CD spectroscopy suggested that (?)- 4 possesses the (1R,4S)-configuration. This absolute configuration was confirmed by single-crystal X-ray diffraction of the derivative (+)-(1R,4R)-3-(1,3-dithian-2-ylidene)-1-azabicyclo [2.2.1]-heptane ((+)- 5 ).