216,100 research outputs found

    Monomial Testing and Applications

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    In this paper, we devise two algorithms for the problem of testing qq-monomials of degree kk in any multivariate polynomial represented by a circuit, regardless of the primality of qq. One is an O(2k)O^*(2^k) time randomized algorithm. The other is an O(12.8k)O^*(12.8^k) time deterministic algorithm for the same qq-monomial testing problem but requiring the polynomials to be represented by tree-like circuits. Several applications of qq-monomial testing are also given, including a deterministic O(12.8mk)O^*(12.8^{mk}) upper bound for the mm-set kk-packing problem.Comment: 17 pages, 4 figures, submitted FAW-AAIM 2013. arXiv admin note: substantial text overlap with arXiv:1302.5898; and text overlap with arXiv:1007.2675, arXiv:1007.2678, arXiv:1007.2673 by other author

    The Cohomology of Transitive Lie Algebroids

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    For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{1*})=H^1_{deR}(M,F_0). Here p^{1*} is the lift of H^1(\huaA,F) to its counterpart over the universal covering space of M and H^1_{deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids.Comment: 17pages, no figure