3,336 research outputs found
Information complexity of the AND function in the two-Party, and multiparty settings
In a recent breakthrough paper [M. Braverman, A. Garg, D. Pankratov, and O.
Weinstein, From information to exact communication, STOC'13] Braverman et al.
developed a local characterization for the zero-error information complexity in
the two party model, and used it to compute the exact internal and external
information complexity of the 2-bit AND function, which was then applied to
determine the exact asymptotic of randomized communication complexity of the
set disjointness problem.
In this article, we extend their results on AND function to the multi-party
number-in-hand model by proving that the generalization of their protocol has
optimal internal and external information cost for certain distributions. Our
proof has new components, and in particular it fixes some minor gaps in the
proof of Braverman et al
On computational complexity of Siegel Julia sets
It has been previously shown by two of the authors that some polynomial Julia
sets are algorithmically impossible to draw with arbitrary magnification. On
the other hand, for a large class of examples the problem of drawing a picture
has polynomial complexity. In this paper we demonstrate the existence of
computable quadratic Julia sets whose computational complexity is arbitrarily
high.Comment: Updated version, to appear in Commun. Math. Phy
On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
Intersection cohomology of Drinfeld's compactifications
Let be a smooth complete curve, be a reductive group and
a parabolic.
Following Drinfeld, one defines a compactification \widetilde{\on{Bun}}_P
of the moduli stack of -bundles on .
The present paper is concerned with the explicit description of the
Intersection Cohomology sheaf of \widetilde{\on{Bun}}_P. The description is
given in terms of the combinatorics of the Langlands dual Lie algebra
.Comment: An erratum adde
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating
4-dimensional super-symmetric gauge theory for a gauge group G with certain
2-dimensional conformal field theory. This conjecture implies the existence of
certain structures on the (equivariant) intersection cohomology of the
Uhlenbeck partial compactification of the moduli space of framed G-bundles on
P^2. More precisely, it predicts the existence of an action of the
corresponding W-algebra on the above cohomology, satisfying certain properties.
We propose a "finite analog" of the (above corollary of the) AGT conjecture.
Namely, we replace the Uhlenbeck space with the space of based quasi-maps from
P^1 to any partial flag variety G/P of G and conjecture that its equivariant
intersection cohomology carries an action of the finite W-algebra U(g,e)
associated with the principal nilpotent element in the Lie algebra of the Levi
subgroup of P; this action is expected to satisfy some list of natural
properties. This conjecture generalizes the main result of arXiv:math/0401409
when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the
works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of
certain shifted Yangians.Comment: minor change
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