A numerical investigation of HELP inequalities

Recent work on the Hardy Everitt Littlewood and Pölya (HELP) inequality using numerical techniques is presented and analysed further. New techniques are used to integrate the highly oscillatory solutions that restricted the range of problems covered in earlier publications.     

Direct measurement of methane hydrate composition along the hydrate equilibrium boundary

The composition of methane hydrate, namely n(w) for CH4.n(w)H2O, was directly measured along the hydrate equilibrium boundary under conditions of excess methane gas. Pressure and temperature conditions ranged from 1.9 to 9.7 MPa and 263 to 285 K. Within experimental error, there is no change in hydrate composition with increasing pressure along the equilibrium boundary, but n(w) may show a slight systematic decrease away from this boundary. A hydrate stoichiometry of n(w) = 5.81-6.10 H2O describes the entire range of measured values, with an average composition of CH4.5.99(+/-0.07)H2O along the equilibrium boundary. These results, consistent with previously measured values, are discussed with respect to the widely ranging values obtained by thermodynamic analysis. The relatively constant composition of methane hydrate over the geologically relevant pressure and temperature range investigated suggests that in situ methane hydrate compositions may be estimated with some confidence.     

Aspects of topology, genus and isomerism in closed 3-valent networks

Recent work on toroidal and Klein bottle fullerenes is reported, including, in particular, attempts to generate useful visual models with the NiceGraph program (part of the Vega package). Some strengths and weaknesses of this modelling software are mentioned. The so‐far‐unanswered question of what ratio of small radius (r) to large radius (R) might be preferred by a geometric realisation of a toroidal network, and broad categories of isomerism are briefly discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

cis- and trans-stereoselective epoxidation of N-protected 2-cyclohexen-1-ylamines

The first systematic study of the cis and trans stereoselectivity in the m-CPBA epoxidation of N-protected cyclic allylic amines has been completed. Mono-N-protected systems gave epoxides with cis stereochemistry (amides are better cis directors than sulfonamides or carbamates) whereas di-N-protected systems gave trans-epoxides (TsNBoc protection gave complete trans stereoselectivity). [structure: see text]     

A spectrophotometric study of o,o'-dihydroxyazobenzene and its chelates with aluminum,gallium and indium

The composition of the chelates of aluminum, gallium and indium with o,o'-dihydroxyazobenzene has been established in acidic alcoholic -water mixtures to be pH dependent by means of spcctrophotometric measurements. Only chelates with a 1 : 1 and 2. : 1 ratio of o,o'-dlihydroxyazobenzene to metal are formed. The first acid dissociation constant of o,o'-dlihydroxyazobenzene and the formulas and stability constants for -the chelates were determined, The stability of the chelates increased in tle order indium, aluminum, gallium.     

Intramolecular general acid catalysis of phosphate transfer. nucleophilic attack by oxyanions on the PO3 2- group

Phosphate transfer from the 8-dimethylammonium-naphthyl-1-phosphate monoanion 4m to water and to a range of nucleophiles shows general acid catalysis by the neighboring NH(+) group, through the strong intramolecular hydrogen bond. Reactivity is insensitive to the charge on the nucleophile, so that fluoride and oxyanions displace dimethylaminonaphthol from the PO3(2-) group as effectively as do amines of the same basicity. Reactivity is (predictably) relatively insensitive to the basicity of the nucleophile and to the alpha-effect. A strong intramolecular hydrogen bond is present in the product, but also in the reactant, as evidenced by major perturbations in the pK(a)'s of the phosphate and dimethylamino groups, to 3.94 and 9.31, respectively, and by ab initio calculations. Rate accelerations are of the order of 10(6)-fold despite this stabilization: the strength of the hydrogen bond is evidently significantly enhanced in the transition state. The evidence suggests that it also depends remarkably strongly on the degree of ionization of the reacting phosphate group and will be significantly reduced for the neutral PO(OH)2 group. Thus, the hydrolysis of the substrate cation 4+ shows a correspondingly greater, >10(8)-fold acceleration.