Determining the effect of asymmetric data on the variogram. I. Underlying asymmetry

Matheron's usual variogram estimator can result in unreliable variograms when data are strongly asymmetric or skewed. Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. This paper examines the effects of underlying asymmetry on the variogram and on the accuracy of prediction, and the second one examines the effects arising from outliers. Standard geostatistical texts suggest ways of dealing with underlying asymmetry; however, this is based on informed intuition rather than detailed investigation. To determine whether the methods generally used to deal with underlying asymmetry are appropriate, the effects of different coefficients of skewness on the shape of the experimental variogram and on the model parameters were investigated. Simulated annealing was used to create normally distributed random fields of different size from variograms with different nugget:sill ratios. These data were then modified to give different degrees of asymmetry and the experimental variogram was computed in each case. The effects of standard data transformations on the form of the variogram were also investigated. Cross-validation was used to assess quantitatively the performance of the different variogram models for kriging. The results showed that the shape of the variogram was affected by the degree of asymmetry, and that the effect increased as the size of data set decreased. Transformations of the data were more effective in reducing the skewness coefficient in the larger sets of data. Cross-validation confirmed that variogram models from transformed data were more suitable for kriging than were those from the raw asymmetric data. The results of this study have implications for the 'standard best practice' in dealing with asymmetry in data for geostatistical analyses. (C) 2007 Elsevier Ltd. All rights reserved.

Determining the effect of asymmetric data on the variogram. II. Outliers

Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. The first paper of this series examined the effects of the former on the variogram and this paper examines the effects of asymmetry arising from outliers. Simulated annealing was used to create normally distributed random fields of different size that are realizations of known processes described by variograms with different nugget:sill ratios. These primary data sets were then contaminated with randomly located and spatially aggregated outliers from a secondary process to produce different degrees of asymmetry. Experimental variograms were computed from these data by Matheron's estimator and by three robust estimators. The effects of standard data transformations on the coefficient of skewness and on the variogram were also investigated. Cross-validation was used to assess the performance of models fitted to experimental variograms computed from a range of data contaminated by outliers for kriging. The results showed that where skewness was caused by outliers the variograms retained their general shape, but showed an increase in the nugget and sill variances and nugget:sill ratios. This effect was only slightly more for the smallest data set than for the two larger data sets and there was little difference between the results for the latter. Overall, the effect of size of data set was small for all analyses. The nugget:sill ratio showed a consistent decrease after transformation to both square roots and logarithms; the decrease was generally larger for the latter, however. Aggregated outliers had different effects on the variogram shape from those that were randomly located, and this also depended on whether they were aggregated near to the edge or the centre of the field. The results of cross-validation showed that the robust estimators and the removal of outliers were the most effective ways of dealing with outliers for variogram estimation and kriging. (C) 2007 Elsevier Ltd. All rights reserved.

The two-dimensional anharmonic oscillator III: the CCC bending mode of C3O2

Newly observed data on the rotational constants of carbon suboxide in excited vibrational states of the low-wavenumber bending vibration ν7 have been successfully interpreted in terms of the two-dimensional anharmonic oscillator wavefunctions associated with this vibration. By combining these results with published infrared and Raman spectra the vibrational assignment has been extended and a refined bending potential for ν7 has been derived: this has a minimum at a bending angle of about 24° at the central C atom, with an energy maximum at the linear configuration some 23 cm−1 above the minimum. From similar data on the combination and hot bands of ν7 with ν4 (1587 cm−1) and ν2 (786 cm−1) the effective ν7 bending potential has also been determined in the one-quantum excited states of ν4 and ν2. The effective ν7 potential shows significant changes from the ground vibrational state; the central hump in the ν7 potential surface is increased to about 50 cm−1 in the v4 = 1 state, and decreased to about 1 cm−1 in the v2 = 1 state. In the light of these results vibrational assignments are suggested for most of the observed bands in the infrared and Raman spectra of C3O2.

Carbon suboxide: the vibrational dependence of the v7 bending potential function

Data on the vibrational energy levels and rotational constants of carbon suboxide for the low-wavenumber bending mode ν7 are reviewed, in the ground-state manifold, and in the ν2-, ν3-, ν4-, and ν2 + ν4-state manifolds. Following the procedure developed by Duckett, Mills, and Robiette [J. Mol. Spectrosc. 63, 249 (1976)] the data have been inverted to give the effective bending potential in ν7 for each of these five states. Values are obtained for various other parameters in the effective vibration-rotation Hamiltonian. The potential and rotational constants in ν2 + ν4 are given to a close approximation by linear extrapolation from the ground state through the ν2 and ν4 states.

The synthesis of chiral N-heterocyclic carbene–borane and –diorganoborane complexes and their use in the asymmetric reduction of ketones

Chiral N-heterocyclic carbene–borane complexes have been synthesised, and have been shown to reduce ketones with Lewis acid promotion. Chiral N-heterocyclic carbene–borane and –diorganoborane complexes can reduce ketones with enantioselectivities up to 75% and 85% ee, respectively.

Total synthesis of (+/-) maculalactone A, maculalactone B and maculalactone C and the determination of the absolute configuration of natural (+) maculalactone A by asymmetric synthesis

Maculalactones A, B and C from the marine cyanobacterium Kyrtuthrix maculans are amongst the only compounds based on the tribenzylbutyrolactone skeleton known in nature and (+) maculalactone A from the natural source possesses significant biological activity against various marine herbivores and marine settlers. We now report a concise synthesis of racemic maculalactone A in five steps from inexpensive starting materials. Maculalactones B and C were synthesized by a minor modification to this procedure, and the synthetic design also permitted an asymmetric synthesis of maculalactone A to be achieved in around 85% ee. The (+) and (-) enantiomers of maculalactone A were assigned, respectively, to the S and R configurations on the basis of the chiral selectivity expected for catecholborane reduction of an unsymmetrical ketone in the presence of Corey's oxazoborolidine catalyst. Surprisingly, it appeared that natural (+) maculalactone A was biosynthesized in K. maculans in a partially racemic form, comprising ca. 90-95% of the (S) enantiomer and 5-10% of its (R) enantiomer. Coincidentally therefore, the percentage enantiomeric excess of the product obtained from asymmetric synthesis almost exactly matched that found in nature. (C) 2004 Elsevier Ltd. All rights reserved.

Asymmetric synthesis of (1R,2S,3R)-3-methylcispentacin and (1S,2S,3R)-3-methyltranspentacin by kinetic resolution of tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate

Conjugate addition of lithium dibenzylamide to tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate occurs with high levels of stereocontrol, with preferential addition of lithium dibenzylamide to the face of the cyclic alpha,beta-unsaturated acceptor anti- to the 3-methyl substituent. High levels of enantiorecognition are observed between tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate and an excess of lithium (+/-)-N-benzyl-N-alpha-methylbenzylamide (10 eq.) (E > 140) in their mutual kinetic resolution, while the kinetic resolution of tert-butyl (+/-)-3-methylcyclopentene-1-carboxylate with lithium (S)-N-benzyl-N-alpha-methylbenzylamide proceeds to give, at 51% conversion, tert-butyl (1R, 2S, 3R,alphaS)-3-methyl-2-N-benzyl-N-alpha-methylbenzylaminocyclopentane-1-c arboxylate consistent with E > 130, and in 39% yield and 99 +/- 0.5% de after purification. Subsequent deprotection by hydrogenolysis and ester hydrolysis gives (1R, 2S, 3R)-3-methylcispentacin in > 98% de and 98 +/- 1% ee. Selective epimerisation of tert-butyl (1R, 2S, 3R, alphaS)-3-methyl-2-N- benzyl-N-alpha-methylbenzylaminocyclopentane-1-carboxylate by treatment with (KOBu)-Bu-t in (BuOH)-Bu-t gives tert-butyl (1S, 2S, 3R, alphaS)-3-methyl-2-N-benzyl-N-alpha-methylbenzylaminocyclopentane-1-carb oxylate in quantitative yield and in > 98% de, with subsequent deprotection by hydrogenolysis and ester hydrolysis giving (1S, 2S, 3R)-3-methyltranspentacin hydrochloride in > 98% de and 97 +/- 1% ee.

Bending, twisting, and breaking CuCN chains to produce framework materials: the reactions of CuCN with alkali-metal halides

The reactions of the low-temperature polymorph of copper(I) cyanide (LT-CuCN) with concentrated aqueous alkali-metal halide solutions have been investigated. At room temperature, KX (X = Br and I) and CsX (X = Cl, Br, and I) produce the addition products K[Cu-2(CN)(2)Br](H2O)-H-. (I), K-3[Cu-6(CN)(6)I-3](.)2H(2)O (II), Cs[Cu-3(CN)(3)Cl] (III), Cs[Cu-3(CN)(3)Br] (IV), and Cs-2[Cu-4(CN)(4)I-2](H2O)-H-. (V), with 3-D frameworks in which the -(CuCN)- chains present in CuCN persist. No reaction occurs, however, with NaX (X = Cl, Br, I) or KCl. The addition compounds, I-V, reconvert to CuCN when washed. Both low- and high-temperature polymorphs of CuCN (LT- and HT-CuCN) are produced, except in the case of Cs[Cu-3(CN)(3)Cl] (III), which converts only to LT-CuCN. Heating similar AX-CuCN reaction mixtures under hydrothermal conditions at 453 K for 1 day produces single crystals of I-V suitable for structure determination. Under these more forcing conditions, reactions also occur with NaX (X = Cl, Br, I) and KCl. NaBr and KCl cause some conversion of LT-CuCN into HT-CuCN, while NaCl and NaI, respectively, react to form the mixed-valence Cu(I)/Cu(II) compounds [Cu-II(OH2)(4)][Cu-4(I)(CN)(6)], a known phase, and [Cu-II(OH2)(4)][Cu-4(I)(CN)(4)I-2] (VI), a 3-D framework, which contains infinite -(CuCN)- chains. After 3 days of heating under hydrothermal conditions, the reaction between KI and CuCN produces [Cu-II(OH2)(4)][Cu-2(I)(CN)I-2](2) (VII), in which the CuCN chains are broken into single Cu-CN-Cu units, which in turn are linked into chains via iodine atoms and then into layers via long Cu-C and Cu-Cu interactions.

Argon predissociation spectroscopy of the OH-center dot H2O and Cl-center dot H2O complexes in the 1000-1900 cm(-1) region: Intramolecular bending transitions and the search for the shared-proton fundamental in the hydroxide monohydrate

We present argon predissociation vibrational spectra of the OH-.H2O and Cl-.H2O complexes in the 1000-1900 cm(-1) energy range, far below the OH stretching region reported in previous studies. This extension allows us to explore the fundamental transitions of the intramolecular bending vibrations associated with the water molecule, as well as that of the shared proton inferred from previous assignments of overtones in the higher energy region. Although the water bending fundamental in the Cl-.H2O spectrum is in very good agreement with expectations, the OH-.H2O spectrum is quite different than anticipated, being dominated by a strong feature at 1090 cm(-1). New full-diniensionality calculations of the OH-.H2O vibrational level structure using diffusion Monte Carlo and the VSCF/CI methods indicate this band arises from excitation of the shared proton.

Stereodivergent approach to the asymmetric synthesis of bacillariolides: A formal synthesis of ent-bacillariolide II

Asymmetric synthesis of densely functionalized bicyclic frameworks for entry into bacillariolides I/III and ent-bacillariolide II is reported. The key features are ring-closing metathesis of a pair of diastereomerically related dienes obtained through a stereodivergent route from a R-(+)-glyceraldehyde derivative, transformation of a nonstereoselective cyclopentene ester enolate alkylation process to a completely stereoselective one through alkylation of a bulky ester enolate with a bulky electrophile, and a remote silyloxymethyl group directed epoxidation.

Asymmetric synthesis of arylalanines via asymmetric aza-darzens (ADZ) reaction

(R)-3-Arylalanines may be prepared in high enantiomeric purity from N-dpp imines by a four-step reaction sequence involving asymmetric aza-Darzens reaction, dephosphinylation, hydrogenolysis and hydrolysis. The amino acids thus obtained were of >95% enantiomeric purity.

Convenient route to enantiopure aryl cyclopentanes via Diels-Alder reaction of asymmetric dienes. Total synthesis of (+)-herbertene and (+)-cuparene

A general route for the synthesis of highly substituted aryl cyclopentanes has been developed involving Diels-Alder reaction of asymmetric dienes prepared from (+)-camphoric acid followed by aromatization of the resulting cyclohexene derivatives. Employing this protocol enantiospecific synthesis of (+)-herbertene and (+)-cuparene has been accomplished. (C) 2003 Elsevier Science Ltd. All rights reserved.