4,646 research outputs found
When is a Schubert variety Gorenstein?
A (normal) variety is Gorenstein if it is Cohen-Macualay and its canonical
sheaf is a line bundle. This property, which measures the ``pathology'' of the
singularities of a variety, is thus stronger than Cohen-Macualayness, but is
also weaker than smoothness. We determine which Schubert varieties are
Gorenstein in terms of a combinatorial characterization using generalized
pattern avoidance conditions. We also give an explicit description as a line
bundle of the canonical sheaf of a Gorenstein Schubert variety.Comment: 15 pages, geometric characterization of Gorensteinness added; final
version to appear in Adv. Mat
Critique of Hirsch's citation index: a combinatorial Fermi problem
The h-index was introduced by the physicist J.E. Hirsch in 2005 as measure of
a researcher's productivity. We consider the "combinatorial Fermi problem" of
estimating h given the citation count. Using the Euler-Gauss identity for
integer partitions, we compute confidence intervals. An asymptotic theorem
about Durfee squares, due to E.R. Canfield-S. Corteel-C.D. Savage from 1998, is
reinterpreted as the rule of thumb h=0.54 x (citations)^{1/2}. We compare these
intervals and the rule of thumb to empirical data (primarily using
mathematicians).Comment: 10 pages + 3 page appendix; 2 figure
Counting magic squares in quasi-polynomial time
We present a randomized algorithm, which, given positive integers n and t and
a real number 0< epsilon <1, computes the number Sigma(n, t) of n x n
non-negative integer matrices (magic squares) with the row and column sums
equal to t within relative error epsilon. The computational complexity of the
algorithm is polynomial in 1/epsilon and quasi-polynomial in N=nt, that is, of
the order N^{log N}. A simplified version of the algorithm works in time
polynomial in 1/epsilon and N and estimates Sigma(n,t) within a factor of
N^{log N}. This simplified version has been implemented. We present results of
the implementation, state some conjectures, and discuss possible
generalizations.Comment: 30 page
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