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

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 $G$-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

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

Compatible metrics on a manifold and non-local bi-Hamiltonian structures

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropriate curvature conditions one may define a curved pencil of compatible metrics and these give rise to an associated non-local bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling and Manin equipped with an invariant metric. In this paper the geometry supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasi-homogeneity assumptions one arrives at a so-called weak \F-manifold - a curved version of a Frobenius manifold (which is not, in general, an F-manifold). A submanifold theory is also developed.Comment: 17 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

Introducing instrumentation and data acquisition to mechanical engineering students using LabVIEW

For several years, LabVIEW has been used within the Department of Mechanical Engineering at the University of Strathclyde as the basis for introducing the basic concepts and practice of data acquisition, and more generally, instrumentation, to postgraduate engineering students and undergraduate project students. The objectives of introducing LabVIEW within the curriculum were to expose students to instrumentation and experimental analysis, and to create courseware that could be used flexibly for a range of students. It was also important that staff time for laboratory work be kept to manageable levels. A course module was developed which allows engineering students with very little or no previous knowledge of instrumentation or programming to become acquainted with the basics of programming, experimentation and data acquisition. The basic course structure has been used to teach both undergraduates and postgraduates as well as laboratory technical staff. The paper describes the objectives of the use of LabVIEW for teaching, the structure of the module developed, and the response of students who have been subjected to the course, and how it is intended to expand the delivery to greater student numbers