546 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
A Judd illusion in far-aiming: evidence of a contribution to action by vision for perception
The present study addresses the role of vision for perception in determining the location of a target in far-aiming. Participants (N = 12) slid a disk toward a distant target embedded in illusory Judd figures. Additionally, in a perception task, participants indicated when a moving pointer reached the midpoint of the Judd figures. The number of hits, the number of misses to the left and to the right of the target, the sliding error (in mm) and perceptual judgment error (in mm) served as dependent variables. Results showed an illusory bias in sliding, the magnitude of which was comparable to the bias in the perception of target location. The determination of target location in far-aiming is thus based on relative metrics. We argue that vision for perception sets the boundary constraints for action and that within these constraints vision for action autonomously controls movement execution, but alternative accounts are discussed as well
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
QM/MM simulations as an assay for carbapenemase activity in class A β-lactamases
Carbapenemases are distinguished from carbapenem-inhibited β-lactamases with a protocol involving QM/MM free energy simulations of acyl–enzyme deacylation, requiring only the enzyme 3D structure as input.</p
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