Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

Constrained Hamiltonian Dynamics

This chapter focuses on autonomous geometrical mechanics, using the language of symplectic geometry. It discusses manifolds (including Kähler manifolds, Riemannian manifolds and Poisson manifolds), tangent bundles, cotangent bundles, vector fields, the Poincaré–Cartan 1-form and Darboux’s theorem. It covers symplectic transforms, the Marsden–Weinstein symplectic quotient, presymplectic and symplectic 2-forms, almost symplectic structures, symplectic leaves and foliation. It also discusses contact structures, musical isomorphisms and Arnold’s theorem, as well as integral invariants, Nambu structures, the Nambu bracket and the Lagrange bracket. It describes Poisson bi-vector fields, Poisson structures, the Lie–Poisson bracket and the Lie–Poisson reduction, as well as Lie algebra, the Lie bracket and Lie algebra homomorphisms. Other topics include Casimir functions, momentum maps, the Euler–Poincaré equation, fibre derivatives and the geodesic equation. The chapter concludes by looking at deformation quantisation of the Poisson algebra, using the Moyal bracket and C*-algebras to develop a quantum physics.

Singular reduction of Nambu–Poisson manifolds

The version of Marsden–Ratiu Poisson reduction theorem for Nambu–Poisson manifolds by a regular foliation have been studied by Ibáñez et al. In this paper, we show that this reduction procedure can be extended to the singular case. Under a suitable notion of Hamiltonian flow on the reduced space, we show that a set of Hamiltonians on a Nambu–Poisson manifold can also be reduced.

De-correlation stretch filtering approach for effective Poisson reduction in galaxy Images

Noise reduction is one of the most important processes to enhance the quality of images. This paper proposes a statistical filter, the decorrelation stretch filter for the reduction of Poisson noise that occurs frequently in galaxy images. The primary purpose of decorrelation stretch is visual enhancement. Decorrstretch is applied to the three band images but can also work on arbitrary number of bands. This filter enhances the color separation of an image with significant band-band correlation. Effectiveness of the proposed filter is compared on the basis of Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE).