Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

The alpha-helix folds more rapidly at the C-terminus than at the N-terminus

The helix-coil dynamics of different sections of an alpha-helical model peptide were observed separately by nanosecond temperature jump experiments with IR detection on a series of isotopically labeled peptides. The results show that the helix-coil dynamics of the alpha-helical C-terminus are faster than those of the N-terminus.     

Simultaneous determination of gallium and aluminium in biological samples by conventional luminescence and derivative synchronous fluorescence spectrometry

The simultaneous determination of gallium and aluminium by using conventional fluorimetry and derivative synchronous fluorescence spectrometry has been studied. These determinations are based on the formation of fluorescent complexes of gallium and aluminium with salicylaldehyde carbohydrazone (SACH). In the conventional method, two samples are analysed under different analytical conditions, and the results are evaluated by solving a system of two simultaneous equations. In the derivative synchronous method (at pH = 2.6, in an ethanol-water medium containing 72% of ethanol), the following conditions are used: a constant wavelength difference of 20 nm between the monochromator settings, a time-constant of 1.5 sec, a scan-speed of 120 nm/min, and a derivative wavelength difference of 10 nm; gallium can be determined in the range 7-38 ng/ml, and aluminium between 6 and 45 ng/ml. The synchronous method shows more advantages, and has been used in the determination of both metal ions in diverse biological samples (animal tissues and human serum) with good results.     

Evaluation of different quantitative approaches for the determination of noneasily ionizable molecules by different atmospheric pressure interfaces used in liquid chromatography tandem mass spectrometry: Abamectin as case of study

The liquid chromatography tandem mass spectroscopy residue determination of compounds without any acidic or basic centers such as abamectin has been investigated. Several approaches regarding the interface used and adduct formation have been compared. The low acidity of the hydroxyl groups only made deprotonation feasible using the atmospheric pressure chemical ionization (APCI) interface. To obtain sufficient sensitivity for residue analysis, the Ion Sabre APCI interface was necessary. However, the sensitivity attained was lower than for monitoring adducts in positive ion mode. Using electrospray ionization, different adducts with Na+, NH4+, and Li+ were tested and compared. The best results were obtained for the ammoniated adduct in electrospray ionization (ESI) because of its high sensitivity and the presence of several product ions with similar abundance. The highest sensitivity was reached using an in-source fragment as precursor ion, leading to a limit of detection (LOD) of 2 microg/L with low relative standard deviation. The relatively high abundance of other transitions allowed abamectin confirmation at concentrations close to the LOD (6 microg/L). Alkali ions were found to be a suitable alternative to determine and confirm abamectin at residue levels. The [M + Na]+ also presented various product ions with similar abundance, which allowed confirmation at LOD levels. However, this LOD was found to be almost four times higher than with [M + NH4]+ because of the poor sensitivity of the transitions obtained. Although the use of Li+ facilitated the fragmentation of the adduct [M + Li]+, with similar sensitivity to [M + NH4]+, this fragmentation preferentially generated only one product ion, which did not allow confirmation at concentration levels lower than 15 microg/L. The use of APCI for monitoring adducts was also feasible, but with less sensitivity. The sensitivity increased with the Ion Sabre APCI, although it was still five times lower than with ESI. Other adduct formers such as Co2+ and Ni2+ also were tested with unsatisfactory results.     

Liquid chromatography and tandem mass spectrometry: a powerful approach for the sensitive and rapid multiclass determination of pesticides and transformation products in water

In this paper we will show the results of our research on the direct simultaneous determination of multi-class pesticides and transformation products with different polarities and acid-base properties by applying an on-line trace enrichment coupled to the chromatographic system supplied with electrospray interface (SPE-LC-MS/MS method). The specific chromatographic separation allows the correct determination of almost fifty compounds (37 pesticides and 10 transformation products) using very low sample volume and very little sample handling. Recoveries between 70-120% were obtained for all compounds in drinking and groundwater, meanwhile in surface water 44 compounds were correctly quantified. Relative standard deviations lower than 15% were obtained for all compounds. Even at the lowest concentration level tested (25 ng L(-1)) 40 compounds presented satisfactory recoveries and repeatability. The use of methanol as organic modifier and the increase of injection volume are also studied. The applicability of the developed method to a monitoring programme is demonstrated by applying it to the analysis of hundreds of samples.     

Highly Enantioselective Construction of Tricyclic Derivatives by the Desymmetrization of Cyclohexadienones

The asymmetric synthesis of tricyclic compounds by the desymmetrization of cyclohexadienones is presented. The reaction tolerated a large variety of substituents at different positions of the cyclohexadienone, and heterocyclic rings of different sizes were accessible. Density functional theory calculations showed that the reaction proceeds through an asynchronous [4+2] cycloaddition.     

Locating a metro line in a historical city centre: application to Sevilla

When constructing a metro alignment under a historical city centre, it is important to generate a cost-effective path while maintaining a minimum distance between the alignment and historical buildings. This paper describes a simple methodology for generating a set of good alternative solutions. It is based on the use of Voronoi diagrams. The method was applied to data from the city of Sevilla.     

Quantification of testosterone undecanoate in human hair by liquid chromatography–tandem mass spectrometry

Testosterone undecanoate (T‐C11) can be used by athletes in order to improve performance. After oral intake, T‐C11 is rapidly metabolized, hampering discrimination between exogenous and endogenous testosterone. A possible alternative is to detect the intact ester in hair. A method based on liquid chromatography–tandem mass spectrometry was developed for the determination of T‐C11 in hair. The sample procedure consisted of digestion of 200 mg of pulverized hair with tris(2‐carboxyethyl)phosphine hydrochloride and liquid–liquid extraction with n‐pentane. Several parameters such as the mobile phase, the ionization source and the washing step were optimized. The method was validated at different spiked levels obtaining satisfactory values for accuracy (between 92 and 102%) with relative standard deviations lower than 7% and a limit of detection of 0.2 ng/g. The applicability of the method was checked by the analysis of three samples from patients using T‐C11. A peak for the analyte was detected in all samples with concentrations between 0.4 and 8.4 ng/g. Copyright © 2009 John Wiley & Sons, Ltd.     

Testosterone metabolism revisited: discovery of new metabolites

The metabolism of testosterone is revisited. Four previously unreported metabolites were detected in urine after hydrolysis with KOH using a liquid chromatography–tandem mass spectrometry method and precursor ion scan mode. The metabolites were characterized by a product ion scan obtained with accurate mass measurements. Androsta-4,6-dien-3,17-dione, androsta-1,4-dien-3,17-dione, 17-hydroxy-androsta-4,6-dien-3-one and 15-androsten-3,17-dione were proposed as feasible structures for these metabolites on the basis of the mass spectrometry data. The proposed structures were confirmed by analysis of synthetic reference compounds. Only 15-androsten-3,17-dione could not be confirmed, owing to the lack of a commercially available standard. That all four compounds are testosterone metabolites was confirmed by the qualitative analysis of several urine samples collected before and after administration of testosterone undecanoate. The metabolite androsta-1,4-dien-3,17-dione has a structure analogous to that of the exogenous anabolic steroid boldenone. Specific transitions for boldenone and its metabolite 17β-hydroxy-5β-androst-1-en-3-one were also monitored. Both compounds were also detected after KOH treatment, suggesting that this metabolic pathway is involved in the endogenous detection of boldenone previously reported by several authors.