Poisson traces,D-modules,and symplectic resolutions

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.     

Nonclassical oxygen atom transfer reactions of oxomolybdenum(VI) bis(catecholate)

Mechanistic studies indicate that the oxomolybdenum(vi) bis(3,5-di-tert-butylcatecholate) fragment deoxygenates pyridine-N-oxides in a reaction where the oxygen is delivered to molybdenum but the electrons for substrate reduction are drawn from the bound catecholate ligands, forming 3,5-di-tert-butyl-1,2-benzoquinone.     

Aerosol retrievals from channel-1 and -2 AVHRR radiances: Long-term trends updated and revisited

The nominal Global Aerosol Climatology Project (GACP) record of aerosol optical thickness (AOT) and Ångström exponent (AE) over the oceans is extended by 6 months to cover the period from August 1981 through December 2005. The most recent 4-year segment reveals no significant short-term tendencies in globally and hemispherically averaged AOTs and AEs. This finding is consistent with contemporaneous MODIS and MISR results and the accumulating evidence of a gradual transition from global brightening to global dimming. We also analyze the retrieval implications of allowing the imaginary part of the aerosols refractive index Im(m) to change over the duration of the GACP record. Our sensitivity study shows that increasing Im(m) from 0.003 during the 4-year pre-Pinatubo period up to 0.007 during the most recent 4-year segment of GACP data eliminates the previously identified long-term decreasing AOT trend. Should this long-term trend in Im(m) be real then it would cause the global absorption AOT over the oceans to more than double and the global single-scattering albedo to decrease from ～0.95 to ～0.88. Such changes could make tropospheric aerosols significant contributors to the recent surface temperature increase.     

Infrared spectra of homogeneous and heterogeneous proton-bound dimers in the gas phase

Infrared multiphoton dissociation spectra of three homogeneous and two heterogeneous proton-bound dimers were recorded in the gas phase. Comparison of the experimental infrared spectra recorded in the fingerprint region of the proton-bound dimers with spectra predicted by electronic structure calculations shows that all modes which are observed contain motion of the proton oscillating between the two monomers. The O-H-O asymmetric stretch for the homogeneous dimers is shown to occur at around 800 cm-1. As expected, the O-H-O asymmetric stretching modes for the heterogeneous proton-bound dimers are observed to shift to significantly higher energy with respect to those for the homogeneous proton-bound dimers due to the asymmetry of the O-H-O moeity. This shift is shown to be predictable from the difference in proton affinities between the two monomers. Density functional predictions of the infrared spectra based on the harmonic oscillator model are demonstrated to predict the observed spectra of the homogeneous proton-bound dimers with reasonable accuracy. Calculations of the structure and infrared spectrum of protonated diglyme at the B3LYP/6-31+G** level and basis also agree well with an infrared spectrum recorded previously. For both heterogeneous proton-bound dimers, however, the predicted spectra are blue-shifted with respect to experiment.     

Stereoisomerism in polyoxometalates: structural and spectroscopic studies of bis(malate)-functionalized cluster systems

The tetrameric macrocycle [(P(mu-NtBu))2(1,4-(NH)2C6H4)]4, obtained from the reaction of the phosphazane dimer [ClP(mu-NtBu)]2 with p-phenylenediamine, has an unusual folded conformation in the solid state and contains a roughly tetrahedral arrangement of endo N-H groups for the potential coordination of anions.     

Structural polarity induced by cooperative hydrogen bonding and lone-pair alignment in the molecular uranyl iodate Na2[UO2(IO3)4(H2O)

Na2[UO2(IO3)4(H2O)] has been synthesized under mild hydrothermal conditions. Its structure consists of Na+ cations and [UO2(IO3)4(H2O)](2-) anions. The [UO2(IO3)4(H2O)](2-) anions are formed from the coordination of a nearly linear uranyl, UO2(2+), cation by four monodentate IO(3-) anions and a coordinating water molecule to yield a pentagonal bipyramidal environment around the uranium center. The water molecules form intermolecular hydrogen bonds with the terminal oxo atoms of neighboring [UO2(IO3)4(H2O)](2-) anions to yield one-dimensional chains that extend down the b axis. There are two crystallographically unique iodate anions in the structure of Na2[UO2(IO3)4(H2O)]. One of these anions is aligned so that the lone-pair of electrons is also directed along the b axis. The overall structure is therefore polar, owing to the cooperative alignment of both the hydrogen bonds and the lone-pair of electrons on iodate. The polarity of the monoclinic space group C2 (a = 11.3810(12) A, b = 8.0547(8) A, c = 7.6515(8) A, beta = 90.102(2) degrees , Z = 2, T = 193 K) found for this compound is consistent with the structure. Second-harmonic generation of 532 nm light from a 1064 nm laser source yields a response of approximately 16x alpha-SiO2.     

Syntheses, structure, magnetism, and optical properties of the interlanthanide sulfides δ-Ln2−xLuxS3 (Ln=Ce, Pr, Nd)

δ-Ln_2−xLu_xS₃ (Ln=Ce, Pr, Nd; x=0.67-0.71) compounds have been synthesized through the reaction of elemental rare-earth metals and S using a Sb₂S₃ flux at 1000 °C. These compounds are isotypic with CeTmS₃, which has a complex three-dimensional structure. It includes four larger Ln³⁺ sites in eight- and nine-coordinate environments, two disordered seven-coordinate Ln³⁺/Lu³⁺ positions, and two six-coordinate Lu³⁺ ions. The structure is constructed from one-dimensional chains of LnS_n (n=6-9) polyhedra that extend along the b-axis. These polyhedra share faces or edges with two neighbors within the chains, while in the [ac] plane they share edges and corners with other chains. Least square refinements gave rise to the formulas of δ-Ce_1.30Lu_0.70S₃, δ-Pr_1.29Lu_0.71S₃ and δ-Nd_1.33Lu_0.67S₃, which are consistent with the EDX analysis and magnetic susceptibility data. δ-Ln_2−xLu_xS₃ (Ln=Ce, Pr, Nd; x=0.67-0.71) show no evidence of magnetic ordering down to 5 K. Optical properties measurements show that the band gaps for δ-Ce_1.30Lu_0.70S₃, δ-Pr_1.29Lu_0.71S₃, and δ-Nd_1.33Lu_0.67S₃ are 1.25, 1.38, and 1.50 eV, respectively. Crystallographic data: δ-Ce_1.30Lu_0.70S₃, monoclinic, space group P2₁/m, a=11.0186(7), b=3.9796(3), c=21.6562(15) Å, β=101.6860(10), V=929.93(11), Z=8; δ-Pr_1.29Lu_0.71S₃, monoclinic, space group P2₁/m, a=10.9623(10), b=3.9497(4), c=21.5165(19) Å, β=101.579(2), V=912.66(15), Z=8; δ-Nd_1.33Lu_0.67S₃, monoclinic, space group P2₁/m, a=10.9553(7), b=3.9419(3), c=21.4920(15) Å, β=101.5080(10), V=909.47(11), Z=8.     

Formation and photoinduced electron ejection of amminedicyanocuprate(I) and cyano-bridged trinuclear copper(I) complexes in aqueous ammonia solution

UV absorption spectral evidence confirms that a mixed-ligand complex, Cu(CN)(2)(NH(3))(-), is formed in an aqueous solution of KCu(CN)(2) and ammonia. The stepwise stability constant for the reaction, Cu(CN)(2)(-) + NH(3) = Cu(CN)(2)(NH(3))(-), is 2.80 +/- 0.40 in 1 M ionic strength, NaClO(4) medium at 25 degrees C. This amminedicyanocuprate(I) ion readily combines in aqueous solution in a 1:2 and 2:1 molar ratios with Cu(NH(3))(2)(+) to form two trinuclear ionic species, presumably with cyano bridges, with the suggested formulas of Cu(3)(CN)(2)(NH(3))(5)(+) and Cu(3)(CN)(4)(NH(3))(3)(-). The resolved UV absorption spectra of the monomer and two trimers have been determined and exhibit strong bands, presumably metal-ligand charge transfer in nature, in the 200-250-nm region. When solutions of all three complexes absorb a pulse of 266-nm laser light, they photoeject hydrated electrons monophotonically, with quantum yields of 0.41 +/- 0.02, 0.53 +/- 0.02, and 0.50 +/- 0.01 for the monomer, cationic trimer, and anionic trimer, respectively, suggesting that absorption in the charge-transfer-to-solvent bands of these complexes results in an efficient electron ejection process that is enhanced by the existence of a polynuclear structure with cyano bridges. No room-temperature luminescence is observed for these complexes.