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

Neurophysiological studies may provide a misleading picture of how perceptual-motor interactions are coordinated

Neurophysiological measurement techniques like fMRI and TMS are increasingly being used to examine the perceptual-motor processes underpinning the ability to anticipate the actions of others. Crucially, these techniques invariably restrict the experimental task that can be used and consequently limit the degree to which the findings can be generalised. These limitations are discussed based on a recent paper by Tomeo et al. (2012) who sought to examine responses to fooling actions by using TMS on participants who passively observed spliced video clips where bodily information was, and was not, linked to the action outcome. We outline two particular concerns with this approach. First, spliced video clips that show physically impossible actions are unlikely to simulate a "fooling" action. Second, it is difficult to make meaningful inferences about perceptual-motor expertise from experiments where participants cannot move. Taken together, we argue that wider generalisations based on these findings may provide a misunderstanding of the phenomenon such a study is designed to explore. © 2013 D Mann, M Dicks, R Cañal-Bruland, J van der Kamp

The chemical potential of the electron gas on a one dimensional lattice

The chemical potential of the electron gas on a one-dimensional lattice is determined within the discrete Hubbard model. The result will have applications in studies of transport properties of quasi one-dimensional organic conductors such as the Bechgaard salts.Comment: 4 pages,plain TeX,presented at the 9 National Congress of Yugoslav Physicists,held in May 1995.,and published in the proceedings.The author can be contacted at: [email protected]

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

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

The effects of high pressure on the point of no return in simulated penalty kicks

We investigated the effects of high pressure on the point of no return or the minimum time required for a kicker to respond to the goalkeeper's dive in a simulated penalty kick task. The goalkeeper moved to one side with different times available for the participants to direct the ball to the opposite side in low-pressure (acoustically isolated laboratory) and high-pressure situations (with a participative audience). One group of participants showed a significant lengthening of the point of no return under high pressure. With less time available, performance was at chance level. Unexpectedly, in a second group of participants, high pressure caused a qualitative change in which for short times available participants were inclined to aim in the direction of the goalkeeper's move. The distinct effects of high pressure are discussed within attentional control theory to reflect a decreasing efficiency of the goal-driven attentional system, slowing down performance, and a decreasing effectiveness in inhibiting stimulus-driven behavior

Twisted reductions of integrable lattice equations, and their Lax representations

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg–de Vries (KdV) equation, discrete modified KdV equation and the discrete Schwarzian KdV equation. We will also describe a direct method for obtaining Lax representations for the reduced equations