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 $\vartheta$-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

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 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

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

An engineering approach to modelling of dynamic insulation using ESP-r

The use of Dynamic Insulation (DI) can enable recovery of conduction heat loss through a building envelope. This is an active process that allows air to move through the fabric against the temperature gradient. Additionally it promises better indoor air quality, primarily due to filtration properties of the construction material [11]. This paper is concerned with quantifying the energy savings and enhancement of human comfort if this technology is integrated into a building. To ascertain the impact of the technology on whole-building performance, it is necessary to undertake detailed dynamic modelling. A suitable building and plant simulation computer tool (ESP-r) was employed to do this. A technique for modelling the dynamic insulation was developed and validated against known analytical solutions. A full-size test house was then simulated, in the UK climate, with and without DI. Comparative results show that better thermal comfort and energy savings are possible with the use of DI. The results obtained have been translated into suggestions for best practice

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