717 research outputs found
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
-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 -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
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
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
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
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