On the stability of weight spaces of enveloping algebra in prime characteristic

By the result of Dixmier, any weight space of enveloping algebra of Lie algebra L over a field of characteristic 0 is adL stable. In this paper we will show that this result need not be true, if F is replaced by a field of prime characteristic. A condition will be given, so a weight space will be adL stable

Stocator: A High Performance Object Store Connector for Spark

We present Stocator, a high performance object store connector for Apache Spark, that takes advantage of object store semantics. Previous connectors have assumed file system semantics, in particular, achieving fault tolerance and allowing speculative execution by creating temporary files to avoid interference between worker threads executing the same task and then renaming these files. Rename is not a native object store operation; not only is it not atomic, but it is implemented using a costly copy operation and a delete. Instead our connector leverages the inherent atomicity of object creation, and by avoiding the rename paradigm it greatly decreases the number of operations on the object store as well as enabling a much simpler approach to dealing with the eventually consistent semantics typical of object stores. We have implemented Stocator and shared it in open source. Performance testing shows that it is as much as 18 times faster for write intensive workloads and performs as much as 30 times fewer operations on the object store than the legacy Hadoop connectors, reducing costs both for the client and the object storage service provider

ON MAXIMAL SUBFIELDS OF ENVELOPING SKEWFIELDS IN PRIME CHARACTERISTICS

Abstract. As was shown by Schue [6] there always exist two maximal subfields of the enveloping skewfields of a solvable Lie p-algebra, such that one is Galois and the second purely inseparable of exponent 1 over the centre. In this paper we obtain similar results for arbitrary solvable Lie algebras in prime characteristic, and for the Zassenhaus algebras. A key result here is to describe relations between maximal subfields in a polynomial extension of a division ring, and those of the base ring. We also provide a description of the enveloping algebra of the p-envelope of a Lie algebra as a polynomial extension of the smaller enveloping algebra