A quantization procedure based on completely positive maps and Markov operators

We describe $\omega$-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces

On discretization in time in simulations of particulate flows

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases

Optimal Lewenstein-Sanpera Decomposition for some Biparatite Systems

It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of product set) on the separable-entangled boundary, optimal Lewenstein-Sanpera (L-S) decomposition can be obtained via optimization for a generic entangled density matrix. Based on this, We obtain optimal L-S decomposition for some bipartite systems such as $2\otimes 2$ and $2\otimes 3$ Bell decomposable states, generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one parameter and three parameters local operations and classical communications (LOCC), $d\otimes d$ Werner and isotropic states, and a one parameter $3\otimes 3$ state. We also obtain the optimal decomposition for multi partite isotropic state. It is shown that in all $2\otimes 2$ systems considered here the average concurrence of the decomposition is equal to the concurrence. We also show that for some $2\otimes 3$ Bell decomposable states the average concurrence of the decomposition is equal to the lower bound of the concurrence of state presented recently in [Buchleitner et al, quant-ph/0302144], so an exact expression for concurrence of these states is obtained. It is also shown that for $d\otimes d$ isotropic state where decomposition leads to a separable and an entangled pure state, the average I-concurrence of the decomposition is equal to the I-concurrence of the state. Keywords: Quantum entanglement, Optimal Lewenstein-Sanpera decomposition, Concurrence, Bell decomposable states, LOCC} PACS Index: 03.65.UdComment: 31 pages, Late

Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence

Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1

Solar Cell Degradation due to Proton Belt Enhancements During Electric Orbit Raising to GEO

The recent introduction of all‐electric propulsion on geosynchronous satellites enables lower‐cost access to space by replacing chemical propellant. However, the time period required to initially raise the satellite to geostationary orbit (GEO) is around 200 days. During this time the satellite can be exposed to dynamic increases in trapped flux which are challenging to model. To understand the potential penalty of this new technique in terms of radiation exposure, the influence of several key parameters on solar cell degradation during the electric orbit raising period has been investigated. This is achieved by calculating the accumulation of non‐ionising dose through time for a range of approaches. We demonstrate the changes in degradation caused by launching during a long‐lived (100s of days) enhancement in MeV trapped proton flux for three different electric orbit raising scenarios and three different thicknesses of coverglass. Results show that launching in an active environment can increase solar cell degradation due to trapped protons by ~5% before start of service compared with a quiet environment. The crucial energy range for such enhancements in proton flux is 3‐10MeV (depending on shielding). Further changes of a few percent can occur between different trajectories, or when a 50μm change in coverglass thickness is applied

Modeling Field Line Curvature Scattering Loss of 1–10 MeV Protons During Geomagnetic Storms

The proton radiation belt contains high fluxes of adiabatically trapped protons varying in energy from ∼one to hundreds of megaelectron volts (MeV). At large radial distances, magnetospheric field lines become stretched on the nightside of Earth and exhibit a small radius of curvature RC near the equator. This leads protons to undergo field line curvature (FLC) scattering, whereby changes to the first adiabatic invariant accumulate as field strength becomes nonuniform across a gyroorbit. The outer boundary of the proton belt at a given energy corresponds to the range of magnetic L shell over which this transition to nonadiabatic motion takes place, and is sensitive to the occurrence of geomagnetic storms. In this work, we first find expressions for nightside equatorial RC and field strength Be as functions of Dst and L* to fit the TS04 field model. We then apply the Tu et al. (2014, https://doi.org/10.1002/2014ja019864) condition for nonadiabatic onset to solve the outer boundary L*, and refine our expression for RC to achieve agreement with Van Allen Probes observations of 1–50 MeV proton flux over the 2014–2018 era. Finally, we implement this nonadiabatic onset condition into the British Antarctic Survey proton belt model (BAS-PRO) to solve the temporal evolution of proton fluxes at L ≤ 4. Compared with observations, BAS-PRO reproduces storm losses due to FLC scattering, but there is a discrepancy in mid-2017 that suggests a ∼5 MeV proton source not accounted for. Our work sheds light on outer zone proton belt variability at 1–10 MeV and demonstrates a useful tool for real-time forecasting

hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated