Quantum Tunneling in the Wigner Representation

Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one - dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semi-classical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem.Comment: 18 pages, LATEX, no figure

Imagining transitions in old age through the visual matrix method: thinking about what is hard to bear

Dominant discourses of ageing are often confined to what is less painful to think about and therefore idealise or denigrate ageing and later life. We present findings from an exploratory psychosocial study, in a Nordic context into three later-life transitions: from working life to retirement, from mental health to dementia, and from life to death. Because, for some, these topics are hard to bear, and therefore defended against and routinely excluded from everyday awareness, we used a method led by imagery and affect - the Visual Matrix - to elicit participants’ free associative personal and collective imagination. Through analysis of data extracts, on the three transitions, we illustrate oscillations between defending against the challenges of ageing and realism in facing the anxieties it can provoke. A recurring theme includes the finality of individual life and the inter-generational continuity, which together link life and death, hope and despair, separation and connectedness

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject