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 $G$-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 $G$-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 $G$-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 $G$-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