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

Astroglial-axonal interactions during early stages of myelination in mixed cultures using in vitro and ex vivo imaging techniques

<b>Background</b><p></p> Myelination is a very complex process that requires the cross talk between various neural cell types. Previously, using cytosolic or membrane associated GFP tagged neurospheres, we followed the interaction of oligodendrocytes with axons using time-lapse imaging in vitro and ex vivo and demonstrated dynamic changes in cell morphology. In this study we focus on GFP tagged astrocytes differentiated from neurospheres and their interactions with axons.<p></p> <b>Results</b><p></p> We show the close interaction of astrocyte processes with axons and with oligodendrocytes in mixed mouse spinal cord cultures with formation of membrane blebs as previously seen for oligodendrocytes in the same cultures. When GFP-tagged neurospheres were transplanted into the spinal cord of the dysmyelinated shiverer mouse, confirmation of dynamic changes in cell morphology was provided and a prevalence for astrocyte differentiation compared with oligodendroglial differentiation around the injection site. Furthermore, we were able to image GFP tagged neural cells in vivo after transplantation and the cells exhibited similar membrane changes as cells visualised in vitro and ex vivo.<p></p> <b>Conclusion</b><p></p> These data show that astrocytes exhibit dynamic cell process movement and changes in their membrane topography as they interact with axons and oligodendrocytes during the process of myelination, with the first demonstration of bleb formation in astrocytes

The dispersive self-dual Einstein equations and the Toda lattice

The Boyer-Finley equation, or $SU(\infty)$-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionlesslimit of the $2d$-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is studied in this paper. The results are achieved by using a deformation, based on an associative $\star$-product, of the algebra $sdiff(\Sigma^2)$ used in the study of the undeformed, or dispersionless, equations.Comment: 11 pages, LaTeX. To appear: J. Phys.

A note on the relationship between rational and trigonometric solutions of the WDVV equations

Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds. As an application it is shown that certain trigonometric and rational solutions of the WDVV equations are related by such a twisted Legendre transform

IEA annex 58 : full-scale empirical validation of detailed thermal simulation programs

As simulation programs become more widely used for building performance assessment and building regulations compliance, there is a need to ensure that there are good quality empirical datasets which can be used to assess the predictive accuracy of these programs. This paper summarises a detailed experiment carried out on two identical full-scale buildings located at the Fraunhofer IBP test site at Holzkirchen in Germany and the associated modelling of the buildings. The work was undertaken as part of IEA ECB Annex 58 "Reliable building energy performance characterization based on full scale dynamic measurements". The test sequence, applied to the side-by-side validation experiment conducted on the multi-roomed Twin Houses, consisted of periods of constant internal temperatures, a period of pseudo-random heat injections and a free-float period. All boundary and internal conditions were comprehensively monitored. Modelling teams were given details of the buildings and the boundary conditions, and over 20 teams submitted their predictions of the internal conditions which were subsequently compared with measurements. The paper focuses on a sensitivity study carried out to assess the overall prediction uncertainty resulting from the uncertainties in the input parameters, as well as identifying those inputs which had the most influence on predictions. An assessment of the measurement uncertainty is also included

The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied.Comment: 29 pages, LaTe

Hydrodynamic reductions of the heavenly equation

We demonstrate that Pleba\'nski's first heavenly equation decouples in infinitely many ways into a triple of commuting (1+1)-dimensional systems of hydrodynamic type which satisfy the Egorov property. Solving these systems by the generalized hodograph method, one can construct exact solutions of the heavenly equation parametrized by arbitrary functions of a single variable. We discuss explicit examples of hydrodynamic reductions associated with the equations of one-dimensional nonlinear elasticity, linearly degenerate systems and the equations of associativity.Comment: 14 page

Rational Approximate Symmetries of KdV Equation

We construct one-parameter deformation of the Dorfman Hamiltonian operator for the Riemann hierarchy using the quasi-Miura transformation from topological field theory. In this way, one can get the approximately rational symmetries of KdV equation and then investigate its bi-Hamiltonian structure.Comment: 14 pages, no figure

Slow Schroedinger dynamics of gauged vortices

Multivortex dynamics in Manton's Schroedinger--Chern--Simons variant of the Landau-Ginzburg model of thin superconductors is studied within a moduli space approximation. It is shown that the reduced flow on M_N, the N vortex moduli space, is hamiltonian with respect to \omega_{L^2}, the L^2 Kaehler form on \M_N. A purely hamiltonian discussion of the conserved momenta associated with the euclidean symmetry of the model is given, and it is shown that the euclidean action on (M_N,\omega_{L^2}) is not hamiltonian. It is argued that the N=3 flow is integrable in the sense of Liouville. Asymptotic formulae for \omega_{L^2} and the reduced Hamiltonian for large intervortex separation are conjectured. Using these, a qualitative analysis of internal 3-vortex dynamics is given and a spectral stability analysis of certain rotating vortex polygons is performed. Comparison is made with the dynamics of classical fluid point vortices and geostrophic vortices.Comment: 22 pages, 2 figure

On the B\"acklund Transformation for the Moyal Korteweg-de Vries Hierarchy

We study the B\"acklund symmetry for the Moyal Korteweg-de Vries (KdV) hierarchy based on the Kuperschmidt-Wilson Theorem associated with second Gelfand-Dickey structure with respect to the Moyal bracket, which generalizes the result of Adler for the ordinary KdV.Comment: 9 pages, Revte