The ancient Mesopotamian music (tonal) system was first interpreted as ascending (Kilmer, Duchesne-Guillemin, Wulstan, Gurney, Thiel and myself). Criticism of the "descending interpretation" (Krispijn, West, and, more recently, Gurney). No argument for this is valid (transformation of original heptatonic and nomenclature, primitive scales, ancient Greece a.o.). Musical-anthropological constants and nomenclature, psychology in the interpretation of notated sources, and the construction of harps all speak for the "ascending" interpretation. Question of Guerney´s new reading...
Shelah's pcf theory describes a certain structure which must exist if $\aleph _{\omega }$ is strong limit and $2^{\aleph _\omega }>\aleph _{\omega _1}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially ordered sets exist.
In this paper we give a new set of verifiable conditions for the existence of average optimal stationary policies in discrete-time Markov decision processes with Borel spaces and unbounded reward/cost functions. More precisely, we provide another set of conditions, which only consists of a Lyapunov-type condition and the common continuity-compactness conditions. These conditions are imposed on the primitive data of the model of Markov decision processes and thus easy to verify. We also give two examples for which all our conditions are satisfied, but some of conditions in the related literature fail to hold.