1,720,994 research outputs found
Haantjes algebras of classical integrable systems
No full textA tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or ωH manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville–Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post–Winternitz system and of a stationary flow of the KdV hierarchy
A Parallel Space Saving Algorithm For Frequent Items and the Hurwitz zeta distribution
Full text linkWe present a message-passing based parallel version of the Space Saving
algorithm designed to solve the --majority problem. The algorithm determines
in parallel frequent items, i.e., those whose frequency is greater than a given
threshold, and is therefore useful for iceberg queries and many other different
contexts. We apply our algorithm to the detection of frequent items in both
real and synthetic datasets whose probability distribution functions are a
Hurwitz and a Zipf distribution respectively. Also, we compare its parallel
performances and accuracy against a parallel algorithm recently proposed for
merging summaries derived by the Space Saving or Frequent algorithms
Partial separability and symplectic-Haantjes manifolds
Full text linkA theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry.
As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to construct sets of coordinates (called Darboux-Haantjes coordinates) which allow both the partial separability of the associated Hamilton-Jacobi equations and the block-diagonalization of the operators of the corresponding Haantjes algebra.
We also introduce a novel class of Hamiltonian systems, characterized by the existence of a generalized Stäckel matrix, which by construction are partially separable. They widely generalize the known families of partially separable Hamiltonian systems. Our systems can be described in terms of semisimple but non-maximal-rank symplectic-Haantjes manifolds.31 page
Haantjes algebras and diagonalization
Full text linkWe introduce the notion of Haantjes algebra: It consists of an assignment of a familyof operator fields on a differentiable manifold, each of them with vanishing Haantjestorsion. They are also required to satisfy suitable compatibility conditions. Haantjesalgebras naturally generalize several known interesting geometric structures, arising inRiemannian geometry and in the theory of integrable systems. At the same time, aswe will show, they play a crucial role in the theory of diagonalization of operatorson differentiable manifolds. Assuming that the operators of a Haantjes algebra aresemisimple and commute, we shall prove that there exists a set of local coordinateswhere all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonalfor
Higher Haantjes Brackets and Integrability
Full text linkWe propose a new, infinite class of brackets generalizing the
Fr\"olicher--Nijenhuis bracket. This class can be reduced to a family of
generalized Nijenhuis torsions recently introduced. In particular, the Haantjes
bracket, the first example of our construction, is relevant in the
characterization of Haantjes moduli of operators.
We shall also prove that the vanishing of a higher-level Nijenhuis torsion of
a given operator is a sufficient condition for the integrability of its
generalized eigen-distributions. This result (which does not require any
knowledge of the spectral properties of the operator) generalizes the
celebrated Haantjes theorem. The same vanishing condition also guarantees that
the operator can be written, in a local chart, in a block-diagonal form.Comment: 24 pages; some results adde
Haantjes structures for the Jacobi-Calogero model and the Benenti systems
Full text linkIn the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized St ̈ackel systems and, as a particular case, of the quasi-bi- Hamiltonian systems. As an application, we recover the Haantjes manifolds for the rational Calogero model with three particles and for the Benenti systems
Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry
Full text linkWe show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ( ) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra, namely an algebra of (1,1)-tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation.
We shall prove that a multiseparable system admits as many structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures
On Appell sequences of polynomials of Bernoulli and Euler type
No full textAbstractA construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented
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