Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

PROPERTIES OF A DIFFERENTIAL SEQUENCE BASED UPON THE KUMMER-SCHWARZ EQUATION

In this paper, we determine a recursion operator for the Kummer-Schwarz equation, which leads to a sequence with unacceptable singularity properties. A different sequence is devised based upon the relationship between the Kummer-Schwarz equation and the first-order Riccati equation for which a particular generator has been found to give interesting and excellent properties. We examine the elements of this sequence in terms of the usual properties to be investigated – symmetries, singularity properties, integrability, alternate sequence – and provide an explanation of the curious relationship between the results of the singularity analysis and a consideration of the solution of each element obtained by quadratures.

Recursion in SPARQL

The need for recursive queries in the Semantic Web setting is becoming more and more apparent with the emergence of datasets where different pieces of information are connected by complicated patterns. This was acknowledged by the W3C committee by the inclusion of property paths in the SPARQL standard. However, as more data becomes available, it is becoming clear that property paths alone are not enough to capture all recursive queries that the users are interested in, and the literature has already proposed several extensions to allow searching for more complex patterns. We propose a rather different, but simpler approach: add a general purpose recursion operator directly to SPARQL. In this paper we provide a formal syntax and semantics for this proposal, study its theoretical properties, and develop algorithms for evaluating it in practical scenarios. We also show how to implement this extension as a plug-in on top of existing systems, and test its performance on several synthetic and real world datasets, ranging from small graphs, up to the entire Wikidata database.

Darboux transformation and exact solutions for a (2 + 1)-dimensional integrable system of nonlinear evolution equations

In this paper, starting from a spectral problem, we construct a [Formula: see text]-dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation.

Direct Construction of a Bi-Hamiltonian Structure for Cubic Hénon-Heiles Systems

The problem of separating variables in integrable Hamiltonian systems has been extensively studied in the last decades. A recent approach is based on the so called Kowalewski's Conditions used to characterized a Control Matrix \(M\) whose eigenvalues give the desired coordinates. In this paper we calculate directly a second compatible Hamiltonian structure for the cubic Hénon-Heiles systems and in this way we obtain the separation variables as the eigenvalues of a recursion operator \(N\). Finally we re-obtain the Control Matrix \(M\) from \(N\).