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.

Impact of forced-aging process on Madeira Wine flavor

The aim of this study was to determine the optimal temperature and baking time to obtain a Madeira wine considered typical by an expert panel. For this purpose simultaneous descriptive analyses of typical Madeira wines were performed, and seven descriptors were selected: “dried fruit”, “nutty”, “musty”, “baked”, “oak”, “mushroom”, and “brown sugar”. Up to 10 odor-active zones were the most frequently cited by the members of the GC-olfactometry panel as corresponding to the panel’s descriptors. The odor importance of each of the zones reported by the GC-O analysis was ranked by AEDA. Three odor zones were identified as common to both Malvasia and Sercial wines and had retention indices (RI) of 1993 (“brown sugar” and “toasted”), 2151 (“brown sugar”), and 2174 (“nutty”, “driedfruits”);sotolonwasidentifiedasresponsibleforthislastaroma.Severalmoleculeswereselected to be quantified on baked wines on the basis of AEDA results and expected Maillard volatiles, such as sotolon, furfural, 5-methylfurfural, 5-ethoximethylfurfural, methional, and phenylacetaldehyde. It was observed that typicity scores were positively correlated with the concentrations of sotolon and sugar and baking time and negatively with the fermentation length.