Sheaves, local homeomorphisms and local sets as Hilbert locale modules

We give a thorough account of the various equivalent notions for \sheaf" on a locale, namely the separated and complete presheaves, the local home- omorphisms, and the local sets, and to provide a new approach based on quantale modules whereby we see that sheaves can be identi¯ed with certain Hilbert modules in the sense of Paseka. This formulation provides us with an interesting category that has immediate meaningful relations to those of sheaves, local homeomorphisms and local sets. The concept of B-set (local set over the locale B) present in [3] is seen as a simetric idempotent matrix with entries on B, and a map of B-sets as de¯ned in [8] is shown to be also a matrix satisfying some conditions. This gives us useful tools that permit the algebraic manipulation of B-sets. The main result is to show that the existing notions of \sheaf" on a locale B are also equivalent to a new concept what we call a Hilbert module with an Hilbert base. These modules are the projective modules since they are the image of a free module by a idempotent automorphism On the ¯rst chapter, we recall some well known results about partially ordered sets and lattices. On chapter two we introduce the category of Sup-lattices, and the cate- gory of locales, Loc. We describe the adjunction between this category and the category Top of topological spaces whose restriction to spacial locales give us a duality between this category and the category of sober spaces. We ¯nish this chapter with the de¯nitions of module over a quantale and Hilbert Module. Chapter three concerns with various equivalent notions namely: sheaves of sets, local homeomorphisms and local sets (projection matrices with entries on a locale). We ¯nish giving a direct algebraic proof that each local set is isomorphic to a complete local set, whose rows correspond to the singletons. On chapter four we de¯ne B-locale, study open maps and local homeo- morphims. The main new result is on the ¯fth chapter where we de¯ne the Hilbert modules and Hilbert modules with an Hilbert and show this latter concept is equivalent to the previous notions of sheaf over a locale.

An improved and fast UHPLC-PDA methodology for determination of L-ascorbic and dehydroascorbic acids in fruits and vegetables: evaluation of degradation rate during storage

This study provides a versatile validated method to determine the total vitamin C content, as the sum of the contents of L-ascorbic acid (L-AA) and dehydroascorbic acid (DHAA), in several fruits and vegetables and its degradability with storage time. Seven horticultural crops from two different origins were analyzed using an ultrahigh-performance liquid chromatographic–photodiode array (UHPLC-PDA) system, equipped with a new trifunctional high strength silica (100% silica particle) analytical column (100 mm×2.1 mm, 1.7 μm particle size) using 0.1% (v/v) formic acid as mobile phase, in isocratic mode. This new stationary phase, specially designed for polar compounds, overcomes the problems normally encountered in HPLC and is suitable for the analysis of large batches of samples without L-AA degradation. In addition, it proves to be an excellent alternative to conventional C18 columns for the determination of L-AA in fruits and vegetables. The method was fully validated in terms of linearity, detection (LOD) and quantification (LOQ) limits, accuracy, and inter/intraday precision. Validation experiments revealed very good recovery rate of 96.6±4.4% for L-AA and 103.1±4.8 % for total vitamin C, good linearity with r2-values >0.999 within the established concentration range, excellent repeatability (0.5%), and reproducibility (1.6%) values. The LOD of the method was 22 ng/mL whereas the LOQ was 67 ng/mL. It was possible to demonstrate that L-AA and DHAA concentrations in the different horticulture products varied oppositely with time of storage not always affecting the total amount of vitamin C during shelf-life. Locally produced fruits have higher concentrations of vitamin C, compared with imported ones, but vegetables showed the opposite trend. Moreover, this UHPLC-PDA methodology proves to be an improved, simple, and fast approach for determining the total content of vitamin C in various food commodities, with high sensitivity, selectivity, and resolving power within 3 min of run analysis.

Tracking of fatness during childhood, adolescence and young adulthood: a 7-year follow-up study in Madeira Island, Portugal

Aims: Investigating tracking of fatness from childhood to adolescence, early adolescence to young adulthood and late adolescence to young adulthood. Subjects and methods: Participants from the Madeira Growth Study were followed during an average period of 7.2 years. Height, body mass, skin-folds and circumferences were measured, nine health- and performance-related tests were administered and the Baecke questionnaire was used to assess physical activity. Skeletal maturity was estimated using the TW3 method. Results: The prevalence of overweight plus obesity ranged from 8.2–20.0% at baseline and from 20.4–40.0% at followup, in boys. Corresponding percentages for girls were 10.6– 12.0% and 13.2–18.0%. Inter-age correlations for fatness indicators ranged from 0.43–0.77. BMI, waist circumference and sum of skin-folds at 8, 12 and 16-years old were the main predictors of these variables at 15, 19 and 23-years old, respectively. Strength, muscular endurance and aerobic fitness were negatively related to body fatness. Physical activity and maturation were independently associated with adolescent (15 years) and young adult (19 years) fatness. Conclusions: Over 7.2 years, tracking was moderate-to-high for fatness. Variance was explained by fatness indicators and to a small extent by physical fitness, physical activity and maturation.