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.

Effect of the estufagem process on the chemical constituents of Madeira wines

Madeira wine is a product of well-established reputation, whose aroma and flavour is the result of unique combinations. Particularly, its maturation may include estufagem, wherein wine is usually heated at 45 °C for three months. During this period, several chemical changes may occur, so it is essential to assess its impact on the wine. In this sense, the main objective of the thesis was to evaluate the effect estufagem on the chemical constituents of Madeira wine, specifically on those molecules potentially important in the development of its typical features. Firstly, analytical methodologies capable of determining the target compounds, combining precision and reproducibility to execution effectiveness, were developed. Then various monovarietal Madeira wines were analysed during estufagem under standard and overheating conditions in order to assess its effect. The following compounds were evaluated: furans, amino acids, biogenic amines, polyphenols, organic acids and volatile compounds. In addition, the total polyphenolic composition, the antioxidant potential and the colour of these wines were also evaluated. The results show that most constituents change due to the heating process. Particularly, the heating promotes the development of 5-hydroxymethylfurfural (HMF) in sweet wines submitted to estufagem at higher temperatures. Moreover, estufagem provides the decrease of most amino acids, suggesting their involvement in the formation of the bouquet of these wines. Regarding the total polyphenol content and antioxidant potential of these wines they do not seem to be greatly affected by the heating step, however most monomeric polyphenols decrease during this process. The thermal processing of Madeira wines favours the development of the volatile composition, especially of volatiles usually reported as typical aromas of Madeira wines. Finally, it was demonstrated that the thermal degradation of sugars, especially of fructose, promotes the emergence of volatile compounds identified in baked wines.