3,418 research outputs found
Deformations of the Monge/Riemann hierarchy and approximately integrable systems
Dispersive deformations of the Monge equation u_u=uu_x are studied using
ideas originating from topological quantum field theory and the deformation
quantization programme. It is shown that, to a high-order, the symmetries of
the Monge equation may also be appropriately deformed, and that, if they exist
at all orders, they are uniquely determined by the original deformation. This
leads to either a new class of integrable systems or to a rigorous notion of an
approximate integrable system. Quasi-Miura transformations are also constructed
for such deformed equations.Comment: 9 pages LaTe
Differential and Functional Identities for the Elliptic Trilogarithm
When written in terms of -functions, the classical
Frobenius-Stickelberger pseudo-addition formula takes a very simple form.
Generalizations of this functional identity are studied, where the functions
involved are derivatives (including derivatives with respect to the modular
parameter) of the elliptic trilogarithm function introduced by Beilinson and
Levin. A differential identity satisfied by this function is also derived.
These generalized Frobenius-Stickelberger identities play a fundamental role in
the development of elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde
equations of associativity, with the simplest case reducing to the above
mentioned differential identity
Symmetries and Solutions of Getzler's Equation for Coxeter and Extended Affine Weyl Frobenius Manifolds
The G-function associated to the semi-simple Frobenius manifold C^n/W (where
W is a Coxeter group or an extended affine Weyl group) is studied. The general
form of the G function is given in terms of a logarithmic singularity over
caustics in the manifold. The main result in this paper is a universal formula
for the G-function corresponding to the Frobenius manifold C^n/W^(k)(A_{n-1})
where W^(k)(A_{n-1}) is a certain extended affine Weyl group (or, equivalently,
corresponding to the Hurwitz space M_{0;k-1,n-k-1}), together with the general
form of the G-function in terms of data on caustics. Symmetries of the G
function are also studied.Comment: 9 pages, LaTe
A construction of Multidimensional Dubrovin-Novikov Brackets
A method for the construction of classes of examples of multi-dimensional,
multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This
is based on an extension of the original construction of Gelfand and Dorfman
which gave examples of Novikov algebras in terms of structures defined from
commutative, associative algebras. Given such an algebra, the construction
involves only linear algebra
Simple Elliptic Singularities: a note on their G-function
The link between Frobenius manifolds and singularity theory is well known,
with the simplest examples coming from the simple hypersurface singularities.
Associated with any such manifold is a function known as the -function. This
plays a role in the construction of higher-genus terms in various theories. For
the simple singularities the G-function is known explicitly: G=0. The next
class of singularities, the unimodal hypersurface or elliptic hypersurface
singularities consists of three examples,
\widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8 (or equivalently P_8,
X_9,J_10). Using a result of Noumi and Yamada on the flat structure on the
space of versal deformations of these singularities the -function is
explicitly constructed for these three examples. The main property is that the
function depends on only one variable, the marginal (dimensionless) deformation
variable. Other examples are given based on the foldings of known Frobenius
manifolds. Properties of the -function under the action of the modular group
is studied, and applications within the theory of integrable systems are
discussed.Comment: 15 page
A Geometry for Multidimensional Integrable Systems
A deformed differential calculus is developed based on an associative
star-product. In two dimensions the Hamiltonian vector fields model the algebra
of pseudo-differential operator, as used in the theory of integrable systems.
Thus one obtains a geometric description of the operators. A dual theory is
also possible, based on a deformation of differential forms. This calculus is
applied to a number of multidimensional integrable systems, such as the KP
hierarchy, thus obtaining a geometrical description of these systems. The limit
in which the deformation disappears corresponds to taking the dispersionless
limit in these hierarchies.Comment: LaTeX, 29 pages. To be published in J.Geom.Phy
Deformations of dispersionless KdV hierarchies
The obstructions to the existence of a hierarchy of hydrodynamic conservation
laws are studied for a multicomponent dispersionless KdV system. It is shown
that if an underlying algebra is Jordan, then the lowest obstruction vanishes
and that all higher obstructions automatically vanish. Deformations of these
multicomponent dispersionless KdV-type equations are also studied. No new
obstructions appear, and hence the existence of a fully deformed hierarchy
depends on the existence of a single purely hydrodynamic conservation law.Comment: 12 papge
Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations
The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.Comment: 28 pages, LaTe
Bayesian Trace Statistics for the Reduced Rank Regression Model.
Estimation of the reduced rank regression model requires restrictions be imposed upon the model. Two forms of restrictions are commonly used. Earlier Bayesian work relied on the triangular method of identification which imposes an a priori ordering on the variables in the system, however, incorrect ordering of the variables can result in model misspecification. Bayesian estimation of the reduced rank regression model without ordering restrictions was presented in Strachan (1998) and follows the classical approach of Anderson (1951) and Johansen (1998). This method of estimation avoids placing restrictions on the space spanned by the reduced rank relations and simplifies testing of restrictions on that space. In this paper, a method for estimating approximate marginal likelihoods and Bayes factors is presented for this model, using Laplace approximations for integrals. These Bayes factors algebraically resemble the Johansen trace statistic (1995), hence the title. We consider the model with rank r and no restrictions on the reduced rank relations.Reduced rank regression, marginal likelihoods, Bayes factors, Bayesian analysis
Generalized Legendre transformations and symmetries of the WDVV equations
The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would
expect from an integrable system, has many symmetries, both continuous and
discrete. One class - the so-called Legendre transformations - were introduced
by Dubrovin. They are a discrete set of symmetries between the stronger concept
of a Frobenius manifold, and are generated by certain flat vector fields. In
this paper this construction is generalized to the case where the vector field
(called here the Legendre field) is non-flat but satisfies a certain set of
defining equations. One application of this more general theory is to generate
the induced symmetry between almost-dual Frobenius manifolds whose underlying
Frobenius manifolds are related by a Legendre transformation. This also
provides a map between rational and trigonometric solutions of the WDVV
equations.Comment: 23 page
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