Riemann–Hilbert problems, Toeplitz operators and Q-classes

We generalize the notion of Q-classes C(Q1,Q2) , which was introduced in the context of Wiener–Hopf factorization, by considering very general 2 × 2 matrix functions Q1, Q2. This allows us to use a mainly algebraic approach to obtain several equivalent representations for each class, to study the intersections of Q-classes and to explore their close connection with certain non-linear scalar equations. The results are applied to various factorization problems and to the study of Toeplitz operators with symbol in a Q-class. We conclude with a group theoretic interpretation of some of the main results.Fundação para a Ciência e a Tecnologia (FCT/Portugal), through Project PTDC/MAT/121837/2010 and Project Est- C/MAT/UI0013/2011. The first author was also supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems and the second author was also supported by the Centre of Mathematics of the University of Minho through the FEDER Funds Programa Operacional Factores de Competitividade COMPET

Investigation of urinary volatile organic metabolites as potential cancer biomarkers by solid-phase microextraction in combination with gas chromatography-mass spectrometry

BACKGROUND: Non-invasive diagnostic strategies aimed at identifying biomarkers of cancer are of great interest for early cancer detection. Urine is potentially a rich source of volatile organic metabolites (VOMs) that can be used as potential cancer biomarkers. Our aim was to develop a generally reliable, rapid, sensitive, and robust analytical method for screening large numbers of urine samples, resulting in a broad spectrum of native VOMs, as a tool to evaluate the potential of these metabolites in the early diagnosis of cancer. METHODS: To investigate urinary volatile metabolites as potential cancer biomarkers, urine samples from 33 cancer patients (oncological group: 14 leukaemia, 12 colorectal and 7 lymphoma) and 21 healthy (control group, cancer-free) individuals were qualitatively and quantitatively analysed. Dynamic solid-phase microextraction in headspace mode (dHS-SPME) using a carboxenpolydimethylsiloxane (CAR/PDMS) sorbent in combination with GC-qMS-based metabolomics was applied to isolate and identify the volatile metabolites. This method provides a potential non-invasive method for early cancer diagnosis as a first approach. To fulfil this objective, three important dHS-SPME experimental parameters that influence extraction efficiency (fibre coating, extraction time and temperature of sampling) were optimised using a univariate optimisation design. The highest extraction efficiency was obtained when sampling was performed at 501C for 60min using samples with high ionic strengths (17% sodium chloride, wv 1) and under agitation. RESULTS: A total of 82 volatile metabolites belonging to distinct chemical classes were identified in the control and oncological groups. Benzene derivatives, terpenoids and phenols were the most common classes for the oncological group, whereas ketones and sulphur compounds were the main classes that were isolated from the urine headspace of healthy subjects. The results demonstrate that compound concentrations were dramatically different between cancer patients and healthy volunteers. The positive rates of 16 patients among the 82 identified were found to be statistically different (Po0.05). A significant increase in the peak area of 2-methyl3-phenyl-2-propenal, p-cymene, anisole, 4-methyl-phenol and 1,2-dihydro-1,1,6-trimethyl-naphthalene in cancer patients was observed. On average, statistically significant lower abundances of dimethyl disulphide were found in cancer patients. CONCLUSIONS: Gas chromatographic peak areas were submitted to multivariate analysis (principal component analysis and supervised linear discriminant analysis) to visualise clusters within cases and to detect the volatile metabolites that are able to differentiate cancer patients from healthy individuals. Very good discrimination within cancer groups and between cancer and control groups was achieved

Screening of volatile composition from portuguese multifloral honeys using headspace solid-phase microextraction-gas chromatography–quadrupole mass spectrometry

The volatile composition from four types of multifloral Portuguese (produced in Madeira Island) honeys was investigated by a suitable analytical procedure based on dynamic headspace solid-phase microextraction (HS-SPME) followed by thermal desorption gas chromatography–quadrupole mass spectrometry detection (GC–qMS). The performance of five commercially available SPME fibres: 100 μm polydimethylsiloxane, PDMS; 85 μm polyacrylate, PA; 50/30 μm divinylbenzene/carboxen on polydimethylsiloxane, DVB/CAR/PDMS (StableFlex); 75 μm carboxen/polydimethylsiloxane, CAR/PDMS, and 65 μm carbowax/divinylbenzene, CW/DVB; were evaluated and compared. The highest amounts of extract, in terms of the maximum signal obtained for the total volatile composition, were obtained with a DVB/CAR/PDMS coating fibre at 60 °C during an extraction time of 40 min with a constant stirring at 750 rpm, after saturating the sample with NaCl (30%). Using this methodology more than one hundred volatile compounds, belonging to different biosynthetic pathways were identified, including monoterpenols, C13-norisoprenoids, sesquiterpenes, higher alcohols, ethyl esters and fatty acids. The main components of the HS-SPME samples of honey were in average ethanol, hotrienol, benzeneacetaldehyde, furfural, trans-linalool oxide and 1,3-dihydroxy-2-propanone

Headspace solid-phase microextraction-gas chromatography-quadrupole mass spectrometric methodology for the establishment of the volatile composition of Passiflora fruit species

Dynamic headspace solid-phase microextraction (HS-SPME) followed by thermal desorption gas chromatography-quadrupole mass spectrometry analysis (GC-qMS), was used to investigate the aroma profile of different species of passion fruit samples. The performance of five commercially available SPME fibres: 65 μm polydimethylsiloxane/divinylbenzene, PDMS/DVB; 100 μm polydimethylsiloxane, PDMS; 85 μm polyacrylate, PA; 50/30 μm divinylbenzene/carboxen on polydimethylsiloxane, DVB/CAR/PDMS (StableFlex); and 75 μm carboxen/polydimethylsiloxane, CAR/PDMS; was evaluated and compared. Several extraction times and temperature conditions were also tested to achieve optimum recovery. The SPME fibre coated with 65 μm PDMS/DVB afforded the highest extraction efficiency, when the samples were extracted at 50 °C for 40 min with a constant stirring velocity of 750 rpm, after saturating the sample with NaCl (17%, w/v — 0.2 g). A comparison among different passion fruit species has been established in terms of qualitative and semi-quantitative differences in volatile composition. By using the optimal extraction conditions and GC-qMS it was possible to tentatively identify seventy one different compounds in Passiflora species: 51 volatiles in Passiflora edulis Sims (purple passion fruit), 24 in P. edulis Sims f. flavicarpa (yellow passion fruit) and 21 compounds in Passiflora mollissima (banana passion fruit). It was found that the ethyl esters comprise the largest class of the passion fruit volatiles, including 82.8% in P. edulis variety, 77.4% in P. edulis Sims f. flavicarpa variety and 39.9% in P. mollissima. The semi-quantitative results were then submitted to principal component analysis (PCA) in order to establish relationships between the compounds and the different passion fruit species under investigation

Changes in volatile composition of Madeira wines during their oxidative ageing

The influence of the age in the volatile composition of Madeira wines made with Boal, Malvazia, Sercial and Verdelho varieties and aged in oak barrel during 1, 11 and 25 years old was been studied. For this purpose, the evolution of volatile compounds: higher alcohols, ethyl esters, fatty acids, furan compounds, enolic compounds, γ-lactones, dioxanes and dioxolanes, of the four most utilised varieties were determined using liquid–liquid extraction with dichloromeihane. Octan-3-ol was used as internal standard. The wines made with these varieties showed great differences in sugar content and small variations on pH and alcoholic degree. The results show that during ageing, the concentration of fatty acids ethyl esters, acetates and fatty acids decrease significantly contrarily to the great increase of ethyl esters of diprotic acids. There is a strong correlation between sotolon, 2-furfural, 5-methyl-2-furfural, 5-hydroxymethyl-2-furfural and 5-ethoxymethyl-2-furfural with wine ageing. These findings indicate that these compounds can be used as ageing wine markers. Among the molecules studied, sotolon [3-hydroxy-4,5-dimethyl-2(5H)-furanone] was one of the few molecules present in concentrations above the perception threshold in Madeira wines. 5-Eihoxymethyl-2-furfural formed from 5-hydroxymethyl-2-furfural and 2-furfural, derived from sugars, are also involved in the aroma of sweet fortified white wines aged in oxidative conditions. The sensory properties change significantly after long periods of conservation

Varietal flavour compounds of four grape varieties producing Madeira wines

Boal, Malvasia, Sercial and Verdelho are the main white grape varieties used in Madeira wine production. To estimate the free fraction of varietal aroma compounds of these varieties, 39 samples of musts were analysed to determine their content of monoterpenols and C13 norisoprenoids (terpenoids), using dynamic headspace solid-phase microextraction coupled with gas chromatography–mass spectrometry. The r-values for linearity studies of the analytical method used, varied between 0.977 (nerolidol) and 0.999 (linalool). The repeatability for each compound varied between 2.5% (citronellol) and 11.8% (β-ionone). The mean values from three vintages (1998, 1999 and 2000) confirmed that these musts have differentiated contents of terpenoids. In opposition to Verdelho musts, Malvasia showed the highest free terpenoids content. In order to establish relations between the compounds and the varieties under investigation, principal component analysis and linear discriminant analysis were applied to the data, revealing a good separation and classification power between the four groups as a function of varietal origin

Comparative analysis of the volatile fraction from Annona cherimola Mill. cultivars by solid-phase microextraction and gas chromatography–quadrupole mass spectrometry detection

The analysis of volatile compounds in Funchal, Madeira, Mateus and Perry Vidal cultivars of Annona cherimola Mill. (cherimoya) was carried out by headspace solid-phase microextraction (HS-SPME) combined with gas chromatography–quadrupole mass spectrometry detection (GC–qMSD). HS-SPME technique was optimized in terms of fibre selection, extraction time, extraction temperature and sample amount to reach the best extraction efficiency. The best result was obtained with 2 g of sample, using a divinylbenzene/carboxen/polydimethylsiloxane (DVB/CAR/PDMS) fibre for 30 min at 30 °C under constant magnetic stirring (800 rpm). After optimization of the extraction methodology, all the cherimoya samples were analysed with the best conditions that allowed to identify about 60 volatile compounds. The major compounds identified in the four cherimoya cultivars were methyl butanoate, butyl butanoate, 3-methylbutyl butanoate, 3-methylbutyl 3-methylbutanoate and 5-hydroxymethyl-2-furfural. These compounds represent 69.08 ± 5.22%, 56.56 ± 15.36%, 56.69 ± 9.28% and 71.82 ± 1.29% of the total volatiles for Funchal, Madeira, Mateus and Perry Vidal cultivars, respectively. This study showed that each cherimoya cultivars have 40 common compounds, corresponding to different chemical families, namely terpenes, esters, alcohols, fatty acids and carbonyl compounds and using PCA, the volatile composition in terms of average peak areas, provided a suitable tool to differentiate among the cherimoya cultivars

Heterocyclic acetals in Madeira wines

The maturation of Madeira wines usually involves exposure to relatively high temperatures which affect the aroma and flavour composition leading to the formation of the typical and characteristic bouquet of these wines. The formation of heterocyclic acetals (1,3-dioxanes and 1,3-dioxolanes) was investigated in order to determine levels and for possible use as indicators of wine age. The results show a linear correlation of the investigated acetals with wine age but suggest that the acetalization reaction is not particularly affected by the drastic oxidative conditions observed during maturation