From Monomials to Words to graphs

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal generated by \sigma(I) in the free monoid is finitely generated. Whether there exists an ordering such that is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs.Comment: 27 pages, 2 postscript figures, uses gastex.st

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SODA: A Lease-Based Consistent Distributed File System

We present a new model for the analysis of the load produced by the lease protocol, a protocol wich assures the consistency of cached information in distributed systems. Using this model, we compare the load produced by this protocol and that produced by the protocol adopted by the SPRITE distributed file system -- which also guarantees the consistency of cached information. We show the superiority of the lease protocol under a large range of our model parameter values. We then describe the SODA consistent distributed file system wich uses an extension of the NFS protocol by addition of leases. Details are shown of an implementation of SODA in the LINUX operating system. The example shows that starting up with the NFS code it should not be hard to implement SODA in other systems. Finally, we present some SODA performance evaluation results and compare them with results obtained in a SPRITE protocol simulator. During this research the first author received a Master's scholarship from ..

Free products of units in algebras - Part I: quaternion algebras

Let A be a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units of A to generate a free group. Using the well-known isomorphism between SO(3; R) and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Specialization techniques allow us to find similar free subgroups in skew polynomial rings. A consequence is the following: let kG be the group algebra of a residually (torsion-free nilpotent) group G over a field k whose characteristic is not 2. If x and y are any pair of non-commuting elements of G, and c; d 2 k then 1 + cx and 1 + dy generate a free subgroup of the Malcev-Neumann field of fractions of kG