1,632 research outputs found
Constructive Relationships Between Algebraic Thickness and Normality
We study the relationship between two measures of Boolean functions;
\emph{algebraic thickness} and \emph{normality}. For a function , the
algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero
coefficients in the unique GF(2) polynomial representing , and the normality
is the largest dimension of an affine subspace on which is constant. We
show that for , any function with algebraic thickness
is constant on some affine subspace of dimension
. Furthermore, we give an algorithm
for finding such a subspace. We show that this is at most a factor of
from the best guaranteed, and when restricted to the
technique used, is at most a factor of from the best
guaranteed. We also show that a concrete function, majority, has algebraic
thickness .Comment: Final version published in FCT'201
Integrable hierarchies and the mirror model of local CP1
We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light
of its realization as a two-component reduction of the two-dimensional Toda
hierarchy, and establish new results on its connection to the Gromov-Witten
theory of local CP1. We first of all elaborate on the relation to the Toeplitz
lattice and obtain a neat description of the Lax formulation of the AL system.
We then study the dispersionless limit and rephrase it in terms of a conformal
semisimple Frobenius manifold with non-constant unit, whose properties we
thoroughly analyze. We build on this connection along two main strands. First
of all, we exhibit a manifestly local bi-Hamiltonian structure of the
Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make
precise the relation between this canonical Frobenius structure and the one
that underlies the Gromov-Witten theory of the resolved conifold in the
equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of
"almost duality" of Frobenius manifolds. As a consequence, we obtain a
derivation of genus zero mirror symmetry for local CP1 in terms of a dual
logarithmic Landau-Ginzburg model.Comment: 27 pages, 1 figur
Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed
We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson
brackets of hydrodynamic type vanishes for almost all degrees. This implies the
existence of a full dispersive deformation of a semisimple bihamiltonian
structure of hydrodynamic type starting from any infinitesimal deformation.Comment: 22 pages. v2: corrected typos. v3: small improvements of the
presentation. v4: typos, small improvements in the introduction and the
presentatio
Bihamiltonian cohomology of KdV brackets
Using spectral sequences techniques we compute the bihamiltonian cohomology
groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In
particular this proves a conjecture of Liu and Zhang about the vanishing of
such cohomology groups.Comment: 16 pages. v2: corrected typos, in particular formulas (28), (78
Reductions of the dispersionless 2D Toda hierarchy and their Hamiltonian structures
We study finite-dimensional reductions of the dispersionless 2D Toda
hierarchy showing that the consistency conditions for such reductions are given
by a system of radial Loewner equations. We then construct their Hamiltonian
structures, following an approach proposed by Ferapontov.Comment: 15 page
The bi-Hamiltonian cohomology of a scalar Poisson pencil
We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless
Poisson pencil in a single dependent variable using a spectral sequence method.
As in the KdV case, we obtain that is isomorphic to
for , to for ,
, , , and vanishes otherwise
- âŠ