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.

Low cost non-intrusive home energy monitoring

This thesis presents a low cost non-intrusive home energy monitor built on top of Non-Intrusive Load Monitoring (NILM) concepts and techniques. NILM solutions are already considered low cost alternatives to the big majority of existing commercial energy monitors but the goal here is to make its cost even lower by using a mini netbook as a whole in one solution. The mini netbook is installed in the homes main circuit breaker and computes power consumption by reading current and voltage from the built-in sound card. At the same time, feedback to the users is provided using the 11’’ LCD screen as well as other built-in I/O modules. Our meter is also capable of detecting changes in power and tries to find out which appliance lead to that change and it is being used as part of an eco-feedback platform that was build to study the long terms of energy eco-feedback in individuals. In this thesis the steps that were taken to come up with such a system are presented, from the basics of AC power measurements to the implementation of an event detector and classifier that was used to disaggregate the power load. In the last chapter results from some validation tests that have been performed are presented in order to validate the experiment. It is believed that such a system will not only be important as an energy monitor, but also as an open system than can be easily changed to accommodate and test new or existing nonintrusive load monitoring techniques.