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

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

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

GRACE and TIMI risk scores but not stress imaging predict long-term cardiovascular follow-up in patients with chest pain after a rule-out protocol

Objective To determine the long-term prognostic value of stress imaging and clinical risk scoring for cardiovascular mortality in chest pain patients after ruling out acute coronary syndrome (ACS). Methods A standard rule-out protocol was performed in emergency room patients with a normal or non-diagnostic admission electrocardiogram (ECG) within 6 h of chest 4 pain onset. ACS patients were identified by troponin T, recurrent angina and serial ECG. Dobutamine stress echocardiography (DSE) was performed after ACS was ruled out. Myocardial perfusion scintigraphy (MPS) was performed within 6 months in an outpatient setting according to the physician's discretion. Results 524 patients were included. GRACE and TIMI risk scores were 75 (57-96) and 1 (0-2) in the rule-out ACS group, and 89 (74-107) and 2 (1-3) in the ACS group, respectively (median, interquartile range). Follow-up (median 9.4 (8.9-10.0) years) was complete in 96%. 350 of 379 rule-out ACS patients had an interpretable DSE and 52 patients underwent an MPS. 21 of the rule-out ACS patients (6%) died of a cardiovascular cause compared with 24 (17%) ACS patients (p <0.001). For rule-out ACS patients, C-statistics were 0.829 and 0.803 for the GRACE and TIMI scores. In these patients, DSE and MPS outcome did not predict long-term cardiovascular mortality. In multivariate analysis, known chronic heart failure, ACE inhibitor use, and GRACE score were independent predictors of cardiovascular mortality. Conclusions TIMI and GRACE score but not DSE and MPS are accurate predictors of long-term cardiovascular mortality, even in chest pain patients with a normal or non-diagnostic electrocardiogram undergoing a rule-out protoco

Clean up procedure (GPC) voor de GC bepaling van de antioxidanten BHA, BHT en TBHQ in botervet

Voor het opstellen van een normvoorschrift voor de bepaling van de antioxydanten BHA, BHT en TBHQ in botervet worden door werkgroep E43 van de "International Dairy Federation" methoden geëvalueerd. Uitgangspunt voor deze norm is de gaschromatografische bepaling beschreven in ISO 6463-1982. De wens bestaat echter om gebruik te maken van capillaire gaschromatografie, dit in afwijking van ISO 6463

Molecular dynamics simulations of the temperature-induced unfolding of crambin follow the Arrhenius equation

Molecular dynamics simulations have been used extensively to model the folding and unfolding of proteins. The rates of folding and unfolding should follow the Arrhenius equation over a limited range of temperatures. This study shows that molecular dynamic simulations of the unfolding of crambin between 500K and 560K do follow the Arrhenius equation. They also show that while there is a large amount of variation between the simulations the average values for the rate show a very high degree of correlation

Terrestrial water load and groundwater fluctuation in the Bengal Basin

Groundwater-level fluctuations represent hydraulic responses to changes in groundwater storage due to aquifer recharge and drainage as well as to changes in stress that include water mass loading and unloading above the aquifer surface. The latter ‘poroelastic’ response of confined aquifers is a well-established phenomenon which has been demonstrated in diverse hydrogeological environments but is frequently ignored in assessments of groundwater resources. Here we present high-frequency groundwater measurements over a twelve-month period from the tropical, fluvio-deltaic Bengal Aquifer System (BAS), the largest aquifer in south Asia. The groundwater level fluctuations are dominated by the aquifer poroelastic response to changes in terrestrial water loading by processes acting over periods ranging from hours to months; the effects of groundwater flow are subordinate. Our measurements represent the first direct, quantitative identification of loading effects on groundwater levels in the BAS. Our analysis highlights the potential limitations of hydrogeological analyses which ignore loading effects in this environment. We also demonstrate the potential for employing poroelastic responses in the BAS and across other tropical fluvio-deltaic regions as a direct, in-situ measure of changes in terrestrial water storage to complement analyses from the Gravity and Climate Experiment (GRACE) mission but at much higher resolution