Resolution of Singularities for a Class of Hilbert Modules

Let M be the completion of the polynomial ring C(z) under bar] with respect to some inner product, and for any ideal I subset of C (z) under bar], let I] be the closure of I in M. For a homogeneous ideal I, the joint kernel of the submodule I] subset of M is shown, after imposing some mild conditions on M, to be the linear span of the set of vectors {p(i)(partial derivative/partial derivative(w) over bar (1),...,partial derivative/partial derivative(w) over bar (m)) K-I] (., w)vertical bar(w=0), 1 <= i <= t}, where K-I] is the reproducing kernel for the submodule 2] and p(1),..., p(t) is some minimal ``canonical set of generators'' for the ideal I. The proof includes an algorithm for constructing this canonical set of generators, which is determined uniquely modulo linear relations, for homogeneous ideals. A short proof of the ``Rigidity Theorem'' using the sheaf model for Hilbert modules over polynomial rings is given. We describe, via the monoidal transformation, the construction of a Hermitian holomorphic line bundle for a large class of Hilbert modules of the form I]. We show that the curvature, or even its restriction to the exceptional set, of this line bundle is an invariant for the unitary equivalence class of I]. Several examples are given to illustrate the explicit computation of these invariants.

Synthesis of substituted nitroolefins: a copper catalyzed nitrodecarboxylation of unsaturated carboxylic acids

A novel, mild and convenient method for the nitrodecarboxylation of substituted cinnamic acid derivatives to their nitroolefins is achieved using a catalytic amount of CuCl (10 mol%) and tert-butyl nitrite (2 equiv.) as a nitrating agent in the presence of air. This reaction provides a useful method for the synthesis of beta,beta-disubstituted nitroolefin derivatives, which are generally difficult to access from other conventional methods. Additionally, this reaction is selective as the E-isomer of the acid derivatives furnishes the corresponding E-nitroolefins. One more salient feature of the method is, unlike other methods, no metal nitrates or HNO3 are employed for the transformation.

The problem of clear air turbulence: Changing perspectives in the understanding of the phenomenon

Due to rapid improvements in on-board instrumentation and atmospheric observation systems, in most cases, aircraft are able to steer clear of regions of adverse weather. However, they still encounter unexpected bumpy flight conditions in regions away from storms and clouds. This is the phenomenon of clear air turbulence (CAT), which has been a challenge to our understanding as well as efforts at prediction. While most of such cases result in mild discomfort, a few cases can be violent leading to serious injuries to passengers and damage to the aircraft. The underlying physical mechanisms have been sought to be explained in terms of fluid dynamic instabilities and waves in the atmosphere. The main mechanisms which have been proposed are: (i) Kelvin-Helmholtz instability of shear layers, (ii) waves generated from flow over mountains, (iii) inertia-gravity waves from clouds and other sources, (iv) spontaneous imbalance theory and (v) horizontal vortex tubes. This has also undergone a change over the years. We present an overview of the mechanisms proposed and their implications for prediction.

Nucleosynthesis in the outflows associated with accretion disks of type II collapsars

We investigate nucleosynthesis inside the outflows from gamma-ray burst (GRB) accretion disks formed by the Type II collapsars. In these collapsars, massive stars undergo core collapse to form a proto-neutron star initially, and a mild supernova (SN) explosion is driven. The SN ejecta lack momentum, and subsequently this newly formed neutron star gets transformed to a stellar mass black hole via massive fallback. The hydrodynamics and the nucleosynthesis in these accretion disks have been studied extensively in the past. Several heavy elements are synthesized in the disk, and much of these heavy elements are ejected from the disk via winds and outflows. We study nucleosynthesis in the outflows launched from these disks by using an adiabatic, spherically expanding outflow model, to understand which of these elements thus synthesized in the disk survive in the outflow. While studying this, we find that many new elements like isotopes of titanium, copper, zinc, etc., are present in the outflows. Ni-56 is abundantly synthesized in most of the cases in the outflow, which implies that the outflows from these disks in a majority of cases will lead to an observable SN explosion. It is mainly present when outflow is considered from the He-rich, Ni-56/Fe-54-rich zones of the disks. However, outflow from the Si-rich zone of the disk remains rich in silicon. Although emission lines of many of these heavy elements have been observed in the X-ray afterglows of several GRBs by Chandra, BeppoSAX, XMM-Newton, etc., Swift seems to have not yet detected these lines.

Cross-Hetero-Dehydrogenative Coupling Reaction of Phosphites: A Catalytic Metal-Free Phosphorylation of Amines and Alcohols

Phosphorylation of amines, alcohols, and sulfoximines are accomplished using molecular iodine as a catalyst and H2O2 as the sole oxidant under mild reaction conditions. This method provides an easy route for synthesizing a variety of phosphoramidates, phosphorus triesters and sulfoximine-derived phosphoramidates which are of biological importance.

Regioselective Synthesis of 4-Substituted Indoles via C-H Activation: A Ruthenium Catalyzed Novel Directing Group Strategy

A highly regioselective functionalization of indole at the C-4 position by employing an aldehyde functional group as a directing group, and Ru as a catalyst, under mild reaction conditions (open flask) has been uncovered. This strategy to synthesize 4-substituted indoles is important, as this class of privileged molecules serves as a precursor for ergot alkaloids and related heterocyclic compounds.

Organization of Mn-Clusters in pcu and bcu Networks: Synthesis, Structure, and Properties

Two new anionic inorganic-organic hybrid compounds H3O]Mn-3(mu(3)-OH)(C14H8O6S)(3)(H2O)](DMF)(5), I, and H3O](2)Mn-7(mu(3)-OH)(4)(C14H8O6S)(6)(H2O)(4)](H2O)(2)(DMF)(8), II have been prepared by employing mild solvothennal methods. Both the compounds have three-dimensionally extended structures formed by Mn-6 and Mn-7 clusters, respectively. The connectivity between Mn-6 and Mn-7 clusters and 4,4'-sulfonyldibenzoic acid anions (SDBA(2-)) results in a six connected pcu network in I and an eight connected bcu network in II. The presence of hydronium ion (H-3(O+)) along with the solvent molecules in the channels of both the compounds suggested proton conduction in the solids. Proton conductivity studies gave values of similar to 3 x 10(-4) Omega(-1) cm(-1) 98% relative humidity in both the compounds. The high activation energies indicate a vehicle mechanism in the compounds I and II. Magnetic studies indicate antiferromagnetic behavior in both the compounds.

Spontaneous assembly of iridium nanochain-like structures: surface enhanced Raman scattering activity using visible light

A facile, environmentally friendly approach to synthesize branched Ir nanochain-like structures under mild conditions, using polyfunctional capping molecules in an aqueous medium is reported; the nanostructures exhibit a surface plasmon resonance peak (SPR) in the visible region and serve as an active substrate for surface enhanced Raman scattering studies.

Copper-Catalyzed Oxidative Transformation of Secondary Alcohols to 1,5-Disubstituted Tetrazoles

A mild and convenient oxidative transformation of secondary alcohols to 1,5-disubstituted tetrazoles is uncovered by employing trimethylsilyl azide (TMSN3) as a nitrogen source in the presence of a catalytic amount of copper(II) perchlorate hexahydrate Cu(ClO4)(2)(.)6H(2)O] (5mol%) and 2,3-dichloro-5,6-dicyano-para-benzoquinone (DDQ) (1.2equiv.) as an oxidant. This reaction is performed under ambient conditions and proceeds through CC bond cleavage.

CHLORAL HYDRATE AS A WATER CARRIER FOR THE EFFICIENT DEPROTECTION OF ACETALS, DITHIOACETALS, AND TETRAHYDROPYRANYL ETHERS IN ORGANIC SOLVENTS

The efficient deprotection of several acetals, dithioacetals, and tetrahydropyranyl (THP) ethers under ambient conditions, using chloral hydrate in hexane, is described. Excellent yields were realized for a wide range of both aliphatic and aromatic substrates. The method is characterized by mild conditions (room temperatures or below), simple workup, and the ready availability of chloral hydrate. High chemoselectivity was also observed in the deprotection, acetonides, esters, and amides being unaffected under the reaction conditions. Products were generally purified chromatographically and identified spectrally. These results constitute a novel addition to current methodology involving a widely employed deprotection tactic in organic synthesis. It seems likely that the mechanism of the reaction involves adsorption of the substrate on the surface of the sparingly soluble chloral hydrate.

SELECTIVE ALDEHYDE REDUCTION IN KETOALDEHYDES WITH NaBH4-Na2CO3-H2O AT ROOM TEMPERATURES

A variety of aliphatic and aromatic ketoaldehydes were reduced to the corresponding ketoalcohols with a mixture of sodium borohydride (1.2 equivalents) and sodium carbonate (sixfold molar excess) in water. Reactions were performed at room temperatures over (typically) 2 h, and yields of isolated products generally ranged from 70% to 85%. A biscarbonate-borane complex, (BH3)(2)CO2](2-) 2Na(+), possibly formed from the reagent mixture, is likely the active reductant. The moderated reactivity of this acylborane species would explain the chemoselectivity observed in the reactions. The readily available reagents and the mild aqueous conditions make for ease of operation and environmental compatibility, and make a useful addition to available methodology.

The morphology dependent electrocatalytic activity of Ir nanostructures towards oxygen reduction

Iridium nanostructures with different morphologies are synthesized by a simple, environmentally friendly approach in aqueous media under mild conditions. The morphology dependent electrocatalytic activity of Ir nanochains and nanoparticles towards oxygen reduction reaction (ORR) has been demonstrated in both acidic and alkaline media. Comparative electrochemical studies reveal that nanochains exhibit significantly enhanced ORR activities in both acidic and alkaline media as compared with nanoparticles, as a result of the continuous structure of interconnected particles. The mechanism of oxygen reduction on Ir nanostructures predominantly follows a four-electron pathway in alkaline and acidic solutions. Excellent stability and good selectivity towards methanol tolerance are reported.

Fabrication of large-area PbSe films at the organic-aqueous interface and their near-infrared photoresponse

An organic-aqueous interfacial reaction at room temperature has been employed to synthesize large-area self-assembled films consisting of PbSe single crystallites. The use of the films for the low-cost fabrication of IR-photodetectors has been explored. (111)-oriented single crystallites of PbSe self-assemble to form robust large-area films. The near-infrared photoresponse of the film measured at room temperature showed large responsivity and gain owing to trap-associated mechanisms. Low-cost, mild reaction conditions and tunability of the nature of deposits make the present strategy useful for synthesizing large-area films of functional materials for possible opto-electronic applications.

Useful approach to the synthesis of aryl thio- and selenoglycosides in the presence of rongalite

A simple, mild, and cost effective methodology has been developed for the synthesis of aryl thio-and selenoglycosides from glycosyl halides and diaryl dichalcogenides. Diaryl dichalcogenides undergo reductive cleavage in the presence of rongalite (HOCH2SO2Na) to generate a chalcogenide anion in situ followed by reaction with glycosyl halides to furnish the corresponding aryl thio- and selenoglycosides in excellent yields. Using this protocol, synthesis of 4-methyl-7-thioumbelliferyl-beta-D-cellobioside (MUS-CB), a fluorescent non-hydrolyzable substrate analogue for cellulases has been achieved. (C) 2014 Elsevier Ltd. All rights reserved.

Improved Erlenmeyer Synthesis with 5-Thiazolone and Catalytic Mn(II) Acetate. Crystal Structure Confirmation of the Reaction Stereochemistry

2-Phenylthiazolin-5-one (5, a thioazlactone) condenses with various aldehydes in the presence of the mild base Mn(II) acetate as catalyst in CH2Cl2 solution. This leads to the corresponding Erlenmeyer reaction products (6) in excellent yields in the case of aromatic aldehydes and moderate yields in others. The mildness of the reaction conditions is apparently enabled by the aromaticity of the (putative) intermediate thiazolone anion. The structure and stereochemistry (Z) of the product derived from i-BuCHO was confirmed by single crystal X-ray diffraction. This study overcomes key limitations of the classical Erlenmeyer synthesis and also introduces the relatively nontoxic Mn(II) acetate as a reagent in heterocyclic chemistry.