159,978 research outputs found
WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6
The first author's previous work established Solomon's WDVV-type relations
for Welschinger's invariant curve counts in real symplectic fourfolds by
lifting geometric relations over possibly unorientable morphisms. We apply her
framework to obtain WDVV-style relations for the disk invariants of real
symplectic sixfolds with some symmetry, in particular confirming Alcolado's
prediction for and extending it to other spaces. These relations
reduce the computation of Welschinger's invariants of many real symplectic
sixfolds to invariants in small degrees and provide lower bounds for counts of
real rational curves with positive-dimensional insertions in some cases. In the
case of , our lower bounds fit perfectly with Koll\'ar's
vanishing results.Comment: 70 pages, 1 figure, 1 table; some updates and modifications in
exposition. arXiv admin note: text overlap with arXiv:1809.0891
Linear orderings of random geometric graphs (extended abstract)
In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges
whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a
certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth,
Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that
some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold
with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic
ordering for our layout problems on the class of random geometric graphs.Postprint (published version
Geometric lower bounds for generalized ranks
We revisit a geometric lower bound for Waring rank of polynomials (symmetric
rank of symmetric tensors) of Landsberg and Teitler and generalize it to a
lower bound for rank with respect to arbitrary varieties, improving the bound
given by the "non-Abelian" catalecticants recently introduced by Landsberg and
Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous
polynomials (partially symmetric tensors); a special case is the simultaneous
Waring decomposition problem for a linear system of polynomials. We generalize
the classical Apolarity Lemma to multihomogeneous polynomials and give some
more general statements. Finally we revisit the lower bound of Ranestad and
Schreyer, and again generalize it to multihomogeneous polynomials and some more
general settings.Comment: 43 pages. v2: minor change
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