2,602 research outputs found
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
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
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
Workshop island 3: algebraic aspects of integrability. Introduction to an additional volume of selected papers arising from the conference on algebraic aspects of integrable systems, Island 3, Islay 2007
As did the very first ISLAND workshop, ISLAND 3 took place on the Hebridean island of Islay, providing a beautiful and serene surrounding for the meeting which ran for over four days. Building on the success of the previous meetings, ISLAND 3 saw the largest number (so far) of participants coming from countries all over the world. A complete list can be found below
On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one
The isomonodromic tau-function for the Hurwitz spaces of branched coverings
of genus zero and one are constructed explicitly. Such spaces may be equipped
with the structure of a Frobenius manifold and this introduces a flat
coordinate system on the manifold. The isomonodromic tau-function, and in
particular the associated -function, are rewritten in these coordinates and
an interpretation in terms of the caustics (where the multiplication is not
semisimple) is given.Comment: 18 page
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
Integrated comparative validation tests as an aid for building simulation tool users and developers
Published validation tests developed within major research projects have been an invaluable aid to program developers to check on their programs. This paper sets out how selected ASHRAE Standard 140-2004 and European CEN standards validation tests have been incorporated into the ESP-r simulation program so that they can be easily run by users and also discusses some of the issues associated with compliance checking. Embedding the tests within a simulation program allows program developers to check routinely whether updates to the simulation program have led to significant changes in predictions and to run sensitivity tests to check on the impact of alternative algorithms. Importantly, it also allows other users to undertake the tests to check that their installation is correct and to give them, and their clients, confidence in results. This paper also argues that validation tests should characterize some of the significant heat transfer processes (particularly internal surface convection) in greater detail in order to reduce the acceptance bands for program predictions. This approach is preferred to one in which validation tests are overly prescriptive (e.g., specifying fixed internal convection coefficients), as these do not reflect how programs are used in practice
Variability in wrist-tilt accelerometer based gesture interfaces
In this paper we describe a study that examines human performance in a tilt control targeting task on a PDA. A three-degree of freedom accelerometer attached to the base of the PDA allows users to navigate to the targets by tilting their wrist in different directions. Post hoc analysis of performance data has been used to classify the ease of targeting and variability of movement in the different directions. The results show that there is an increase in variability of motions upwards from the centre, compared to downwards motions. Also the variability in the x axis component of the motion was greater than that in the y axis. This information can be used to guide designers as to the ease of various relative motions, and can be used to reshape the dynamics of the interaction to make each direction equally easy to achieve
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