A diffusion approximation result for two parameter processes

Summary We consider a one-dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so-called Goursat problem. In order to observe the effect of the randomness on the solution we perform a space-time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes. We prove that the solution converges in distribution toward the solution of a two-parameter stochastic differential equation which we identify. The diffusion approximation results for oneparameter processes are well known and well understood. In fact, the solution of the one-parameter analog of the problem we consider here is immediate. Unfortunately, the situation is much more complicated for two-parameter processes and we believe that our result is the first one of its kind.Partially supported by ONR N00014-91-J-1010     

Difference schemes on uniform grids performed by general discrete operators

Our aim is to set the foundations of a discrete vectorial calculus on uniform n-dimensional grids, that can be easily reformulated on general irregular grids. As a key tool we first introduce the notion of tangent space to any grid node. Then we define the concepts of vector field, field of matrices and inner products on the space of grid functions and on the space of vector fields, mimicking the continuous setting. This allows us to obtain the discrete analogous of the basic first order differential operators, gradient and divergence, whose composition define the fundamental second order difference operator. As an application, we show that all difference schemes, with constant coefficients, for first and second order differential operators with constant coefficients can be seen as difference operators of the form for suitable choices of q, and  . In addition, we characterize special properties of the difference scheme, such as consistency, symmetry and positivity in terms of q, and  .     

Thiocarbonyl rhodium complexes with tricyclohexylphosphine and other nitrogen and phosphorus donor ligands

Summary The preparation of the covalent Rh(OCIO₃)(CS)(PCy₃)₂ and Rh(OClO₃)(CS)(PPh₃)(PCy₃) perchlorato complexes is described, These complexes react with mono- or bidentate nitrogen donor ligands to give new cationic complexes of the [Rh(CS)(PCY₃)₂L]ClO4 and [Rh(CS)(PPh,)(PCy₃)L]ClO₄ types,     

Synthesis and Glycosidase Inhibitory Activity of 7‐Deoxycasuarine

Reaction of 1,4‐anhydro‐2,3,5‐tri‐O‐benzyl‐1‐deoxy‐1‐imino‐D ‐arabinitol N‐oxide ( 8 ) with allyl alcohol produced a 3.6 : 1 mixture of the two pyrrolo[1,2‐b]isoxazole derivatives 13 and 14 . The major adduct 13 was converted to 7‐deoxycasuarine ( 7 ), a potent, specific, and competitive inhibitor of amyloglucosidase from Rhizopus mold (see Table).     

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.     

Synthesis, solid-state structure, and bonding analysis of the beryllocenes [Be(C5Me4H)2], [Be(C5Me5)2], and [Be(C5Me5)(C5Me4H)

The beryllocenes [Be(C(5)Me(4)H)(2)] (1), [Be(C(5)Me(5))(2)] (2), and [Be(C(5)Me(5))(C(5)Me(4)H)] (3) have been prepared from BeCl(2) and the appropriate KCp' reagent in toluene/diethyl ether solvent mixtures. The synthesis of 1 is facile (20 degrees C, overnight), but generation of decamethylberyllocene 2 demands high temperatures (ca. 115 degrees C) and extended reaction times (3-4 days). The mixed-ring beryllocene 3 is obtained when the known [(eta(5)-C(5)Me(5))BeCl] is allowed to react with K[C(5)Me(4)H], once more under somewhat forcing conditions (115 degrees C, 36 h). The structures of the three metallocenes have been determined by low-temperature X-ray studies. Both 1 and 3 present eta5/eta1 geometries of the slip-sandwich type, whereas 2 exhibits an almost regular, ferrocene-like, sandwich structure. In the mixed-ring compound 3, C(5)Me(5) is centrally bound to beryllium and the eta(1)-C(5)Me(4)H ring bonds to the metal through the unique CH carbon atom. This is also the binding mode of the eta(1)-ring of 1. To analyze the nature of the bonding in these molecules, theoretical calculations at different levels of theory have been performed on compounds 2 and 3, and a comparison with the bonding in [Be(C(5)H(5))(2)] has been made. As for the latter molecule, energy differences between the eta5/eta5 and the eta5/eta1 structures of 2 are very small, being of the order of a few kcal mol(-1). Constrained space orbital variations (CSOV) calculations show that the covalent character in the bonding is larger for [Be(C(5)Me(5))(2)] than for [Be(C(5)H(5))(2)] due to larger charge delocalization and to increased polarizability of the C(5)Me(5) fragment.     

Preparation and properties of dinitrogen complexes of molybdenum and tungsten with trimethylphosphine as coligand : III. Synthesis and properties of cis-[W(N2)2(PMe3)4], trans-[W(C2H4)2(PMe3)4] and [M(N2)(PMe3)5] (M = Mo,W). The crystal and molecular structure of [Mo(N2)(PMe3)5]

The reduction of [WCl₄(PMe₃)₃] with dispersed sodium, under dinitrogen, gives cis-[W(N₂)₂(PMe₃)₄], while under ethylene trans-[W(C₂H₄)₂(PMe₃)₄] is obtained. The ethylene complex can also be prepared by displacement of the dinitrogen molecules in cis-[W(N₂)₂(PMe₃)₄] by ethylene at room temperature and pressure. Interaction of cis-[M(N₂)₂(PMe₃)₄] complexes (M = Mo, W), with PMe₃, under helium or argon, yields [M(N₂)(PMe₃)₅]. The molybdenum complex crystallizes in the orthorhombic space group Pnma, with a 22.063(6), b 12.106(4), c 9.745(4) Å. The Mo—P distance trans to the dinitrogen ligand (2.483(7) Å) is slightly longer than the average of the other four Mo—P bonds (2.460(5) Å).     

Barreleneiridium(I) complexes. Crystal structures of [Ir(Me3TFB)(η6-C6H4Me2)]ClO4 and [Ir(TFB)(η5-PhNPh2)]BF4·CH2Cl2 (TFB = tetrafluorobenzobarrelene)

A new series of cationic areneiridium(I) complexes of formula [Ir(barrelene)(arene)]⁺ or [Ir(barrelene)(PhNRPh)]⁺ (R= Ph or H) have been synthesized from neutral iridium complexes of the type [IrY(barrelene)]_x (barrelene = Me₃TFB, Y = Cl or OMe (x = 2), Y = acac (x = 1); barrelene = TFB, Y = OMe (x = 2), Y = acac (x = 1)). The crystal structures of [Ir(Me₃TFB)(1,4-C₆H₄Me₂)]ClO₄ and [Ir(TFB)(PhNPh₂)]BF₄·CH₂Cl₂ have been determined by X-ray diffraction. They crystallize in the space groups Pbca and Pna2₁ respectively with lattice constants of 17.6947(11), 15.8072(10), 16.0019(11) Å and 9.8059(2), 20.8097(9), 14.3367(4) Å. Final R factors were 0.063 and 0.042 for the observed data. Both complexes show a staggered arrangement between the arene and the TFB moieties and deviation from planarity of the coordinated arene ligands. In the second complex the IrC and NC distances, the CNC angle, the type of arene puckering, and the spectroscopic data indicate a distortion of the coordinated arene towards a η⁵-coordinated iminocyclohexadienyl form.     

Stereoselective syntheses of 1,4-dideoxy-1,4-imino-octitols and novel tetrahydroxyindolizidines

A new route for the preparation of four new indolizidines, (1R,2S,6S,7S,8aS)- and (1R,2S,6R,7R,8aS)-1,2,6,7-tetrahydroxyindolizidine (30 and 32) and (1S,2R,7S,8S,8aR)- and (1S,2R,7R,8R,8aR)-1,2,7,8-tetrahydroxyindolizidine (44 and 46), is reported. The synthesis is based on Knoevenagel homologation of the readily available enantiomerically pure pyrrolidin-carbaldehydes 13 and 37followed by asymmetric dihydroxylation of the subsequent alkenyl pyrrolidines and cyclization of the corresponding imino-octitols. The new indolizidines and their precursors (imino-octitols 20, 25, 26) and indolizidinones 28a and 28b have been tested for inhibitory activities toward 26 glycosidases. The enzymatic inhibition of trans-7-hydroxy-d-(-)-swainsonine (44) toward alpha-mannosidases is similar to that described for trans-7-hydroxy-l-(+)-swainsonine (11b) toward naringinase (alpha-l-rhamnosidase from Penicillium decumbens).