The staircase method: integrals for periodic reductions of integrable lattice equations

We show, in full generality, that the staircase method provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r<n, then one can introduce q<= 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular 2D lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.Comment: 40 pages, 23 Figure

The Solar Photospheric Nitrogen Abundance: Determination with 3D and 1D Model Atmospheres

We present a new determination of the solar nitrogen abundance making use of 3D hydrodynamical modelling of the solar photosphere, which is more physically motivated than traditional static 1D models. We selected suitable atomic spectral lines, relying on equivalent width measurements already existing in the literature. For atmospheric modelling we used the co 5 bold 3D radiation hydrodynamics code. We investigated the influence of both deviations from local thermodynamic equilibrium (non-LTE effects) and photospheric inhomogeneities (granulation effects) on the resulting abundance. We also compared several atlases of solar flux and centre-disc intensity presently available. As a result of our analysis, the photospheric solar nitrogen abundance is A(N) = 7.86 +/- 0.12.Comment: 6 pages, 4 figure

Isotope Spectroscopy

The measurement of isotopic ratios provides a privileged insight both into nucleosynthesis and into the mechanisms operating in stellar envelopes, such as gravitational settling. In this article, we give a few examples of how isotopic ratios can be determined from high-resolution, high-quality stellar spectra. We consider examples of the lightest elements, H and He, for which the isotopic shifts are very large and easily measurable, and examples of heavier elements for which the determination of isotopic ratios is more difficult. The presence of 6Li in the stellar atmospheres causes a subtle extra depression in the red wing of the 7Li 670.7 nm doublet which can only be detected in spectra of the highest quality. But even with the best spectra, the derived $^6$Li abundance can only be as good as the synthetic spectra used for their interpretation. It is now known that 3D non-LTE modelling of the lithium spectral line profiles is necessary to account properly for the intrinsic line asymmetry, which is produced by convective flows in the atmospheres of cool stars, and can mimic the presence of 6Li. We also discuss briefly the case of the carbon isotopic ratio in metal-poor stars, and provide a new determination of the nickel isotopic ratios in the solar atmosphere.Comment: AIP Thinkshop 10 "High resolution optical spectroscopy", invited talk, AN in pres

Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight functions in the orthogonality condition. In this paper we extend this mechanism to a new class of two-variable orthogonal polynomials where the variables are related via an elliptic curve. This leads to a `Higher order Analogue of the Discrete-time Toda' (HADT) equation for the associated Hankel determinants, together with its Lax pair, which is derived from the relevant recurrence relations for the orthogonal polynomials. In a similar way as the quotient-difference (QD) algorithm is related to the discrete-time Toda equation, a novel quotient-quotient-difference (QQD) scheme is presented for the HADT equation. We show that for both the HADT equation and the QQD scheme, there exists well-posed $s$-periodic initial value problems, for almost all \s\in\Z^2. From the Lax-pairs we furthermore derive invariants for corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page

Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and Discrete Darboux Polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, all rational preserved integrals can be found. We show, in two examples, how to use this method to detect and determine preserved measures and integrals of the considered rational maps.Comment: 8 pages, 1 Figur

CO ro-vibrational lines in HD100546: A search for disc asymmetries and the role of fluorescence

We have studied the emission of CO ro-vibrational lines in the disc around the Herbig Be star HD100546 with the final goal of using these lines as a diagnostic to understand inner disc structure in the context of planet formation. High-resolution IR spectra of CO ro-vibrational emission at eight different position angles were taken with CRIRES at the VLT. From these spectra flux tables, CO ro-vibrational line profiles, and population diagrams were produced. We have investigated variations in the line profile shapes and line strengths as a function of slit position angle. We used the thermochemical disc modelling code ProDiMo based on the chemistry, radiation field, and temperature structure of a previously published model for HD100546. Comparing observations and the model, we investigated the possibility of disc asymmetries, the excitation mechanism (UV fluorescence), the geometry, and physical conditions of the inner disc. The observed CO ro-vibrational lines are largely emitted from the inner rim of the outer disc at 10-13 AU. The line shapes are similar for all v levels and line fluxes from all vibrational levels vary only within one order of magnitude. All line profile asymmetries and variations can be explained with a symmetric disc model to which a slit correction and pointing offset is applied. Because the angular size of the CO emitting region (10-13 AU) and the slit width are comparable the line profiles are very sensitive to the placing of the slit. The model reproduces the line shapes and the fluxes of the v=1-0 lines as well as the spatial extent of the CO ro-vibrational emission. It does not reproduce the observed band ratios of 0.5-0.2 with higher vibrational bands. We find that lower gas volume densities at the surface of the inner rim of the outer disc can make the fluorescence pumping more effcient and reproduce the observed band ratios.Comment: 20 pages, 21 figure

Eigen model as a quantum spin chain: exact dynamics

We map Eigen model of biological evolution [Naturwissenschaften {\bf 58}, 465 (1971)] into a one-dimensional quantum spin model with non-Hermitean Hamiltonian. Based on such a connection, we derive exact relaxation periods for the Eigen model to approach static energy landscape from various initial conditions. We also study a simple case of dynamic fitness function.Comment: 10 pages. Physical Revew E vol. 69, in press (2004

Capitals and commitment. The case of a local learning and employment network.

This article draws on research undertaken with a Local Learning and Employment Network (LLEN) in the state of Victoria, Australia. LLEN are networks that were implemented by the state government in 2001 to undertake community capacity building through which the outcomes of young people aged 15-19 in education, training and employment would be enhanced. In 2008, in the context of an enhanced federal commitment to social inclusion through ‘joining-up’, the Victorian experience provides insights on the implications of such policy initiatives. Drawing on Bourdieu’s discussion of the forms of capital and Granovetter’s notion of the strength of weak ties, I argue that stores of economic, cultural and social capital as outlined by Bourdieu were necessary, but insufficient, for LLEN to achieve the objectives with which they were charged given the failure of government to follow through on the implications of its policies. I argue for a commitment on the part of all stakeholders to realise the potential of ‘joining-up’

Presupposition projection as proof construction

Even though Van der Sandt's presuppositions as anaphora approach is empirically successful, it fails to give a formal account of the interaction between world-knowledge and presuppositions. In this paper, an algorithm is sketched which is based on the idea of presuppositions as anaphora. It improves on this approach by employing a deductive system, Constructive Type Theory (CTT), to get a formal handle on the way world-knowledge influences presupposition projection. In CTT, proofs for expressions are explicitly represented as objects. These objects can be seen as a generalization of DRT's discourse markers. They are useful in dealing with presuppositional phenomena which require world-knowledge, such as Clark's bridging examples and Beaver's conditional presuppositions