Non-commutative complete intersections and the homology of a Shafarevich complex

Abstract

Stochastic resonance in a globally coupled neuronal network has been studied via numerical simulation. The ability of the network to detect a weak (subthreshold) periodic signal can be optimized t o a high signal-to-noise ratio with a long plateau as the noise intensity increases. This may interpret the strong ability of neurons to process information in biological system.

Noncommutative complete intersections

“…A few examples of noncommutative (or quantum) complete intersections have been constructed and studied along the line of factoring out a regular sequence of elements [9,10,42]. Further, different kinds of generalizations of a commutative complete intersection have been proposed during the last fifteen years [20,17,6,23]. Recent work on noncommutative (or twisted) matrix factorizations [13], derived representation schemes [8], noncommutative versions of support varieties and finite generation of the cohomology ring of a Hopf algebra (ideas similar to [7,41]), as well as noncommutative crepant resolutions of commutative schemes [16], advocate for a better understanding of noncommutative complete intersections.…”

Algebras associated to acyclic directed graphs

“…In Section 5 we study the behavior of the Hilbert series under operations on graphs. In particular, we use these results to construct a natural family of noncommutative complete intersections (in the sense of [1], [7]. )…”

“…Recall that according to [7] if V is a graded vector space with graded dimension H(V, z) and R is a graded subspace of the tensor algebra…”

“…Recall that according to [7] if V is a graded vector space with graded dimension H(V, z) and R is a graded subspace of the tensor algebra T (V ) with graded dimension H(R, z), the quotient A = T (V )/ < R > is said to be a noncommutative complete intersection if…”