654 research outputs found
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
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
Towards abundance estimates for jack mackerel in the South Pacific based on acoustic data collected by the commercial vessels
Pelagic trawlers make intensive use of echosounders and therefore could potentially be used as acoustic data collection platforms. This project investigated the possibility of collecting acoustic data and its potential utility to estimate fish stock biomass. The scope of the project was to develop and - when possible - test the tools that would be necessary for large scale data collection from commercial vessels, and investigate the suitability of acoustic data to derive abundance indices
Trees and superintegrable Lotka-Volterra families
To any tree on vertices we associate an -dimensional Lotka-Volterra
system with parameters and prove it is superintegrable, i.e. it admits
functionally independent integrals. We also show how these systems can be
reduced to an ()-dimensional system which is superintegrable and solvable
by quadratures.Comment: 13 pages, 2 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
The Need for Testing—The Exercise Challenge Test to Disentangle Causes of Childhood Exertional Dyspnea
Exertional dyspnea is a common symptom in childhood which can induce avoidance of physical activity, aggravating the original symptom. Common causes of exertional dyspnea are exercise induced bronchoconstriction (EIB), dysfunctional breathing, physical deconditioning and the sensation of dyspnea when reaching the physiological limit. These causes frequently coexist, trigger one another and have overlapping symptoms, which can impede diagnoses and treatment. In the majority of children with exertional dyspnea, EIB is not the cause of symptoms, and in asthmatic children it is often not the only cause. An exercise challenge test (ECT) is a highly specific tool to diagnose EIB and asthma in children. Sensitivity can be increased by simulating real-life environmental circumstances where symptoms occur, such as environmental factors and exercise modality. An ECT reflects daily life symptoms and impairment, and can in an enjoyable way disentangle common causes of exertional dyspnea
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
Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families
We present a method to construct superintegrable -component Lotka-Volterra
systems with parameters. We apply the method to Lotka-Volterra systems
with components for , and present several -dimensional
superintegrable families. The Lotka-Volterra systems are in one-to-one
correspondence with trees on vertices.Comment: 14 pages, 4 figure
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