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

Involutivity of integrals for sine-Gordon, modified KdV and potential KdV maps

Closed form expressions in terms of multi-sums of products have been given in \cite{Tranclosedform, KRQ} of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as so-called $(p,-1)$-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products.Comment: 24 page

Anticipation and visual search behaviour in expert soccer goalkeepers

A novel methodological approach is presented to examine the visual search behaviours employed by expert goalkeepers during simulated penalty kick situations in soccer. Expert soccer goalkeepers were classified as successful or unsuccessful based on their performance on a film-based test of anticipation skill, thereby allowing an intra-group comparison of visual search behaviour on the task. The anticipation test required participants to move a joystick in response to penalty kick situations presented on a large screen. The proportion of penalties saved was assessed as well as the frequency and time of initiation of joystick corrections. Visual search behaviour was examined using a portable eye movement registration system. The successful experts were more accurate in predicting the height and direction of the penalty kick, waited longer before initiating a response and appeared to spend longer periods of time fixating on the non-kicking leg compared with the nonsuccessful experts. © 2005 Taylor & Francis

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

Implicit learning increases shot accuracy of football players when making strategic decisions during penalty kicking

Implicit learning has been proposed to improve athletes’ performance in dual-task situations. Yet, only a few studies tested this with a sports-relevant dual-task. Hence, the current study aimed to compare the effects of implicit and explicit training methods on penalty kicking performance. Twenty skilled football players were divided in two training groups and took part in a practice phase to improve kicking accuracy (i.e., without a goalkeeper) and in a post-test in order to check penalty kick performance (i.e., accuracy including a decision to kick to the side opposite the goalkeeper's dive). Results found that the implicit and explicit training method resulted in similar levels of decision-making, but after implicit training this was achieved with higher kicking accuracy. Additionally, applications for football players and coaches are discussed

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

Path to Actinorhodin:Regio- and Stereoselective Ketone Reduction by a Type II Polyketide Ketoreductase Revealed in Atomistic Detail

In type II polyketide synthases (PKSs), which typically biosynthesize several antibiotic and antitumor compounds, the substrate is a growing polyketide chain, shuttled between individual PKS enzymes, while covalently tethered to an acyl carrier protein (ACP): this requires the ACP interacting with a series of different enzymes in succession. During biosynthesis of the antibiotic actinorhodin, produced by Streptomyces coelicolor, one such key binding event is between an ACP carrying a 16-carbon octaketide chain (actACP) and a ketoreductase (actKR). Once the octaketide is bound inside actKR, it is likely cyclized between C7 and C12 and regioselective reduction of the ketone at C9 occurs: how these elegant chemical and conformational changes are controlled is not yet known. Here, we perform protein-protein docking, protein NMR, and extensive molecular dynamics simulations to reveal a probable mode of association between actACP and actKR; we obtain and analyze a detailed model of the C7-C12-cyclized octaketide within the actKR active site; and we confirm this model through multiscale (QM/MM) reaction simulations of the key ketoreduction step. Molecular dynamics simulations show that the most thermodynamically stable cyclized octaketide isomer (7R,12R) also gives rise to the most reaction competent conformations for ketoreduction. Subsequent reaction simulations show that ketoreduction is stereoselective as well as regioselective, resulting in an S-alcohol. Our simulations further indicate several conserved residues that may be involved in selectivity of C7-12 cyclization and C9 ketoreduction. Detailed insights obtained on ACP-based substrate presentation in type II PKSs can help design ACP-ketoreductase systems with altered regio- or stereoselectivity

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