An Iterated Forcing Extension In Which All Aleph-1 Dense Sets of Reals Are Isomorphic

If κ is an infinite cardinal, then a subset X of the reals R is called κ-dense if between any a \u3c b, both in R, there are exactly κ elements of X. In these terms, a famous result of Cantor says that in every model of set theory all aleph-0-dense sets of reals are isomorphic (to the rationals, Q). This result cannot be directly extended, however, since for κ =aleph-1 there exist models of set theory in which not all aleph-1-dense sets are isomorphic. On the other hand, Baumgartner has shown by the method of iterated forcing that assuming the consistency of set theory, there does exist at least one model of set theory in which all aleph-1-dense subsets are isomorphic. We present here a detailed, yet expository, account of Baumgartner\u27s result and discuss its relevance to the Proper Forcing Axiom of contemporary set theory

Permanent Magnet Assisted Synchronous Reluctance Machine (PMa-SynRM) Design and Performance Analysis for Fan and Pump Applications

One of the major industrial applications of electric machines is driving fans and pumps. According to the pump and fan affinity laws, decreasing the speed of the load can reduce the power consumption on the load significantly. Therefore, a variable speed drive offers reduction in energy consumption of the induction motor driving those types of loads. As an alternative, synchronous machines can be used for this application to get the benefit of higher efficiency. In this work, the performance of an optimized permanent magnet assisted synchronous reluctance machine (PMa-SynRM) with a NEMA standard stator has been studied for fan and pump applications. The effect of using different quantities of the permanent magnet in this machine is studied experimentally. In addition, the performance of the PMa-SynRM is compared with a standard general purpose induction machine for the same loading condition. This work presents the comparison of the energy consumption and performance of the machines under the fan and pump type loads operating on specific typical duty cycles