Natural coordinates for a class of Benenti systems

We present explicit formulas for the coordinates in which the Hamiltonians of the Benenti systems with flat metrics take natural form and the metrics in question are represented by constant diagonal matrices.Comment: LaTeX 2e, 8 p., no figures; extended version with enlarged bibliograph

Cognitive dimensions of public space

The paper presents a new approach to cognitive aspects of public space based on the Bayesian framework for cognition. According to it, cognition is powered by hypothesis-testing brain, constantly minimizing its prediction error. Expectations the brain generates can be analyzed at three different levels of organization: (1) neural implementation, comprising of three distinctive cortical networks, (2) mental computation, consisting of three parts of the Bayes’ rule, and (3) social behavior inside three different social networks. Properly designed the public space can be part of the extended mind of its inhabitants, enhancing or substituting their brains’ activity

Construction of coupled Harry Dym hierarchy and its solutions from St\"ackel systems

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical St\"ackel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding St\"ackel systems.Comment: 16 page

Miura maps for St\"{a}ckel systems

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyperelliptic type. These Miura maps induce Miura maps between St\"{a}ckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for St\"{a}ckel systems

Non-Hamiltonian systems separable by Hamilton-Jacobi method

We show that with every separable calssical Stackel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These system are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.Comment: 20 pages, LaTeX, no figure

Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems

Rational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multi-Hamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical R-matrix approach applied to Poisson algebras. A proof, that Poisson operators constructed near different points of Laurent expansion of Lax functions are equal, is given. All results are illustrated by several examples.Comment: 28 page

Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

A systematic construction of St\"{a}ckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is derived. One Casimir bi-Hamiltonian case is studed in details and in this case, a systematic construction of related hydrodynamic systems in arbitrary coordinates is presented, using a cofactor method and soliton symmetry constraints.Comment: to appear in Journal of Geometry and Physic