41,270 research outputs found
Differential constraints compatible with linearized equations
Differential constraints compatible with the linearized equations of partial
differential equations are examined. Recursion operators are obtained by
integrating the differential constraints
Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula
Starting from the Rodrigues representation of polynomial solutions of the
general hypergeometric-type differential equation complementary polynomials are
constructed using a natural method. Among the key results is a generating
function in closed form leading to short and transparent derivations of
recursion relations and an addition theorem. The complementary polynomials
satisfy a hypergeometric-type differential equation themselves, have a
three-term recursion among others and obey Rodrigues formulas. Applications to
the classical polynomials are given.Comment: 13 pages, no figure
Nijenhuis operator in contact homology and descendant recursion in symplectic field theory
In this paper we investigate the algebraic structure related to a new type of
correlator associated to the moduli spaces of -parametrized curves in
contact homology and rational symplectic field theory. Such correlators are the
natural generalization of the non-equivariant linearized contact homology
differential (after Bourgeois-Oancea) and give rise to an invariant Nijenhuis
(or hereditary) operator (\`a la Magri-Fuchssteiner) in contact homology which
recovers the descendant theory from the primaries. We also sketch how such
structure generalizes to the full SFT Poisson homology algebra to a (graded
symmetric) bivector. The descendant hamiltonians satisfy to recursion
relations, analogous to bihamiltonian recursion, with respect to the pair
formed by the natural Poisson structure in SFT and such bivector. In case the
target manifold is the product stable Hamiltonian structure , with
a symplectic manifold, the recursion coincides with genus topological
recursion relations in the Gromov-Witten theory of .Comment: 30 pages, 3 figure
Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
The exact free energy of matrix model always obeys the Seiberg-Witten (SW)
equations on a complex curve defined by singularities of the quasiclassical
resolvent. The role of SW differential is played by the exact one-point
resolvent. We show that these properties are preserved in generalization of
matrix models to beta-ensembles. However, since the integrability and
Harer-Zagier topological recursion are still unavailable for beta-ensembles, we
need to rely upon the ordinary AMM/EO recursion to evaluate the first terms of
the genus expansion. Consideration in this paper is restricted to the Gaussian
model.Comment: 15 page
The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion
We introduce the boundary length and point spectrum, as a joint
generalization of the boundary length spectrum and boundary point spectrum in
arXiv:1307.0967. We establish by cut-and-join methods that the number of
partial chord diagrams filtered by the boundary length and point spectrum
satisfies a recursion relation, which combined with an initial condition
determines these numbers uniquely. This recursion relation is equivalent to a
second order, non-linear, algebraic partial differential equation for the
generating function of the numbers of partial chord diagrams filtered by the
boundary length and point spectrum.Comment: 16 pages, 6 figure
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