Sudden death of entanglement induced by polarization mode dispersion

We study the decoherence of polarization-entangled photon pairs subject to the effects of polarization mode dispersion, the chief polarization decoherence mechanism in optical fibers. We show that fiber propagation reveals an intriguing interplay between the concepts of entanglement sudden death, decoherence-free subspaces, and nonlocality. We define the boundaries in which entanglement-based quantum communications protocols relying on fiber propagation can be applied.     

Effect of fiber-spinning profile on plug-and-play quantum-key distribution systems

The spin profile of a fiber is usually optimized to reduce transmission impairments caused by polarization-mode dispersion (PMD). In this paper, we show that fiber-optic-based plug-and-play quantum-key distribution systems using polarization modulation and fibers with a spin profile optimal for PMD may suffer from a large Faraday rotation induced by the geomagnetic field. We show that, for periodic spin patterns of small periods, the Faraday rotation is minimum when no spin is applied to the fiber.     

Strong trajectory attractors for dissipative Euler equations

The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R>0 are considered and the existence of a strong trajectory attractor in the space is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method.     

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic Kähler manifold, we define the symplectic Wirtinger number W(X). We show that and the equality is reached if and only if the subvariety is trianalytic, i.e. compatible with the hyperkähler structure on M. For a sequence of immersions of simple holomorphic symplectic manifolds, we show that      

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

A hypercomplex manifold is a manifold equipped with a triple of complex structures I, J, K satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics. We prove a quaternionic analogue of A. D. Aleksandrov and ChernLevine-Nirenberg theorems.     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .