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

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 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

Singularity, complexity, and quasi--integrability of rational mappings

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)--integrability, and in particular its links with their singularities (in the 2--plane). Finally, we describe some of their properties {\it qua\/} dynamical systems, making contact with Arnol'd's notion of complexity, and exemplify remarkable behaviours.Comment: Latex file. 17 pages. To appear in CM

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects in the regular square lattice are discussed for the discrete potential KdV and linear wave equations. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Comment: Corrected version with the assumption of nonsingularity of solutions explicitly state

Baxterization, dynamical systems, and the symmetries of integrability

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to [email protected] and give your postal mail addres

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Cluster mutation-periodic quivers and associated Laurent sequences

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale. Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages, 35 figure

Diagnostic accuracy of endoscopic ultrasonography-guided tissue acquisition prior to resection of pancreatic carcinoma:a nationwide analysis

Introduction: Endoscopic ultrasonography guided tissue acquisition (EUS + TA) is used to provide a tissue diagnosis in patients with suspected pancreatic cancer. Key performance indicators (KPI) for these procedures are rate of adequate sample (RAS) and sensitivity for malignancy (SFM). Aim: assess practice variation regarding KPI of EUS + TA prior to resection of pancreatic carcinoma in the Netherlands. Patients and methods: Results of all EUS + TA prior to resection of pancreatic carcinoma from 2014–2018, were extracted from the national Dutch Pathology Registry (PALGA). Pathology reports were classified as: insufficient for analysis (b1), benign (b2), atypia (b3), neoplastic other (b4), suspected malignant (b5), and malignant (b6). RAS was defined as the proportion of EUS procedures yielding specimen sufficient for analysis. SFM was calculated using a strict definition (malignant only, SFM-b6), and a broader definition (SFM-b5+6). Results: 691 out of 1638 resected patients (42%) underwent preoperative EUS + TA. RAS was 95% (range 89–100%), SFM-b6 was 44% (20–77%), and SFM-b5+6 was 65% (53–90%). All centers met the performance target RAS&gt;85%. Only 9 out of 17 met the performance target SFM-b5+6 &gt; 85%. Conclusion: This nationwide study detected significant practice variation regarding KPI of EUS + TA procedures prior to surgical resection of pancreatic carcinoma. Therefore, quality improvement of EUS + TA is indicated

Diagnostic accuracy of endoscopic ultrasonography-guided tissue acquisition prior to resection of pancreatic carcinoma:a nationwide analysis

Introduction: Endoscopic ultrasonography guided tissue acquisition (EUS + TA) is used to provide a tissue diagnosis in patients with suspected pancreatic cancer. Key performance indicators (KPI) for these procedures are rate of adequate sample (RAS) and sensitivity for malignancy (SFM). Aim: assess practice variation regarding KPI of EUS + TA prior to resection of pancreatic carcinoma in the Netherlands. Patients and methods: Results of all EUS + TA prior to resection of pancreatic carcinoma from 2014–2018, were extracted from the national Dutch Pathology Registry (PALGA). Pathology reports were classified as: insufficient for analysis (b1), benign (b2), atypia (b3), neoplastic other (b4), suspected malignant (b5), and malignant (b6). RAS was defined as the proportion of EUS procedures yielding specimen sufficient for analysis. SFM was calculated using a strict definition (malignant only, SFM-b6), and a broader definition (SFM-b5+6). Results: 691 out of 1638 resected patients (42%) underwent preoperative EUS + TA. RAS was 95% (range 89–100%), SFM-b6 was 44% (20–77%), and SFM-b5+6 was 65% (53–90%). All centers met the performance target RAS&gt;85%. Only 9 out of 17 met the performance target SFM-b5+6 &gt; 85%. Conclusion: This nationwide study detected significant practice variation regarding KPI of EUS + TA procedures prior to surgical resection of pancreatic carcinoma. Therefore, quality improvement of EUS + TA is indicated.</p