Families of classical subgroup separable superintegrable systems

We describe a method for determining a complete set of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates. As examples, we use it to determine complete sets of integrals, polynomial in the momenta, for some families of generalized oscillator and Kepler-Coulomb systems, hence demonstrating their superintegrability. The latter generalizes recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz. Another example is given of a superintegrable system on a non-conformally flat space.Comment: 9 page

Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

This article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. In the first part of the article we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different three-dimensional spaces. We show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is Stäckel equivalent to a system on a constant curvature space. In the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates. We find that there are eight families of these systems

Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields.

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level m vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras

Quantum models related to fouled Hamiltonians of the harmonic oscillator

We study a pair of canonoid (fouled) Hamiltonians of the harmonic oscillator which provide, at the classical level, the same equation of motion as the conventional Hamiltonian. These Hamiltonians, say $K_{1}$ and $K_{2}$, result to be explicitly time-dependent and can be expressed as a formal rotation of two cubic polynomial functions, $H_{1}$ and $H_{2}$, of the canonical variables (q,p). We investigate the role of these fouled Hamiltonians at the quantum level. Adopting a canonical quantization procedure, we construct some quantum models and analyze the related eigenvalue equations. One of these models is described by a Hamiltonian admitting infinite self-adjoint extensions, each of them has a discrete spectrum on the real line. A self-adjoint extension is fixed by choosing the spectral parameter $\epsilon$ of the associated eigenvalue equation equal to zero. The spectral problem is discussed in the context of three different representations. For $\epsilon =0$, the eigenvalue equation is exactly solved in all these representations, in which square-integrable solutions are explicity found. A set of constants of motion corresponding to these quantum models is also obtained. Furthermore, the algebraic structure underlying the quantum models is explored. This turns out to be a nonlinear (quadratic) algebra, which could be applied for the determination of approximate solutions to the eigenvalue equations.Comment: 24 pages, no figures, accepted for publication on JM

Using social media for asynchronous collaboration within collaborative networks

Societal challenges of today (e.g. aging) are complex and often require systemic solutions to be addressed. To address these challenges, various expertise and knowledge are required; in this sense, collaborative network projects have a lot of potential in offering a systemic solution. Design workshops (synchronous collaboration) are often used to achieve progress in such projects. In this paper we introduce asynchronous collaboration, which can occur anytime, anywhere through the use of social media. We have probed Instagram as a ‘ready-made’ social media platform within two collaborative network project case studies. This was done to experiment with asynchronous collaboration and knowledge sharing in addition to design workshops. Both cases were evaluated through focus groups that indicated how social media has the potential to enable actors to cross-field boundaries, inspire each other, and in this way enrich the design process within asynchronous collaboration. Our contribution with this work is two-fold: on the one hand, we aim to inspire and show how collaborative network projects can benefit from asynchronous collaboration in addition to synchronous collaboration. On the other hand, we hope to contribute to the creation of specific social media platforms as tools for supporting asynchronous collaboration within collaborative networks

Using social media for asynchronous collaboration within collaborative networks

Societal challenges of today (e.g. aging) are complex and often require systemic solutions to be addressed. To address these challenges, various expertise and knowledge are required; in this sense, collaborative network projects have a lot of potential in offering a systemic solution. Design workshops (synchronous collaboration) are often used to achieve progress in such projects. In this paper we introduce asynchronous collaboration, which can occur anytime, anywhere through the use of social media. We have probed Instagram as a ‘ready-made’ social media platform within two collaborative network project case studies. This was done to experiment with asynchronous collaboration and knowledge sharing in addition to design workshops. Both cases were evaluated through focus groups that indicated how social media has the potential to enable actors to cross-field boundaries, inspire each other, and in this way enrich the design process within asynchronous collaboration. Our contribution with this work is two-fold: on the one hand, we aim to inspire and show how collaborative network projects can benefit from asynchronous collaboration in addition to synchronous collaboration. On the other hand, we hope to contribute to the creation of specific social media platforms as tools for supporting asynchronous collaboration within collaborative networks