Uncertainty Relations for Angular Momentum

In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the "uncertainty regions'' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r(s), where r(s)/s goes to 1 as s tends to infinity.Comment: 30 pages, 22 figures, 1 cut-out paper model, video abstract available on https://youtu.be/h01pHekcwF

Stability of solutions to chance constrained stochastic programs

Perturbations of convex chance constrained stochastic programs are considered the underlying probability distributions of which are r-concave. Verifiable sufficient conditions are established guaranteeing Hölder continuity properties of solution sets with respect to variations of the original distribution. Examples illustrate the potential, sharpness and limitations of the results

Problem-based optimal scenario generation and reduction in stochastic programming

Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem

On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata [17]. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived (Proposition 3.2). Later the co-derivative formula is used for verifying constraint qualifications and for identifying M-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios

Metric regularity and quantitative stability in stochastic programs with probabilistic constraints

Necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general nonconvex, nonsmooth probabilistic constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to establish a large deviation estimate for solution sets when the probability measure is replaced by empirical ones

The springtail cuticle as a blueprint for omniphobic surfaces

Omniphobic surfaces found in nature have great potential for enabling novel and emerging products and technologies to facilitate the daily life of human societies. One example is the water and even oil-repellent cuticle of springtails (Collembola). The wingless arthropods evolved a highly textured, hierarchically arranged surface pattern that affords mechanical robustness and wetting resistance even at elevated hydrostatic pressures. Springtail cuticle-derived surfaces therefore promise to overcome limitations of lotus-inspired surfaces (low durability, insufficient repellence of low surface tension liquids). In this review, we report on the liquid-repellent natural surfaces of arthropods living in aqueous or temporarily flooded habitats including water-walking insects or water spiders. In particular, we focus on springtails presenting an overview on the cuticular morphology and chemistry and their biological relevance. Based on the obtained liquid repellence of a variety of liquids with remarkable efficiency, the review provides general design criteria for robust omniphobic surfaces. In particular, the resistance against complete wetting and the mechanical stability strongly both depend on the topographical features of the nano- and micropatterned surface. The current understanding of the underlying principles and approaches to their technological implementation are summarized and discussed