18,539 research outputs found
Cohomology of Heisenberg Lie Superalgebras
Suppose the ground field to be algebraically closed and of characteristic
different from and . All Heisenberg Lie superalgebras consist of two
super versions of the Heisenberg Lie algebras, and
with a nonnegative integer and a positive integer. The
space of a "classical" Heisenberg Lie superalgebra is the
direct sum of a superspace with a non-degenerate anti-supersymmetric even
bilinear form and a one-dimensional space of values of this form constituting
the even center. The other super analog of the Heisenberg Lie algebra,
, is constructed by means of a non-degenerate anti-supersymmetric
odd bilinear form with values in the one-dimensional odd center. In this paper,
we study the cohomology of and with
coefficients in the trivial module by using the Hochschild-Serre spectral
sequences relative to a suitable ideal. In characteristic zero case, for any
Heisenberg Lie superalgebra, we determine completely the Betti numbers and
associative superalgebra structure for their cohomology. In characteristic
case, we determine the associative superalgebra structures for the
divided power cohomology of and we also make an attempt to
determine the cohomology of by computing it in a
low-dimensional case.Comment: 19 page
A Web Aggregation Approach for Distributed Randomized PageRank Algorithms
The PageRank algorithm employed at Google assigns a measure of importance to
each web page for rankings in search results. In our recent papers, we have
proposed a distributed randomized approach for this algorithm, where web pages
are treated as agents computing their own PageRank by communicating with linked
pages. This paper builds upon this approach to reduce the computation and
communication loads for the algorithms. In particular, we develop a method to
systematically aggregate the web pages into groups by exploiting the sparsity
inherent in the web. For each group, an aggregated PageRank value is computed,
which can then be distributed among the group members. We provide a distributed
update scheme for the aggregated PageRank along with an analysis on its
convergence properties. The method is especially motivated by results on
singular perturbation techniques for large-scale Markov chains and multi-agent
consensus.Comment: To appear in the IEEE Transactions on Automatic Control, 201
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