Polytopes in all dimensional loop quantum gravity

The Lasserre’s reconstruction algorithm is extended to the D-polytopes with the construction of their shape space. Thus, the areas of d-skeletons $$(1\le d\le D)$$ ( 1 ≤ d ≤ D ) can be expressed as functions of the areas and normal bi-vectors of the (D-1)-faces of D-polytopes. As weak solutions of the simplicity constraints in all dimensional loop quantum gravity, the simple coherent intertwiners are employed to describe semiclassical D-polytopes. New general geometric operators based on D-polytopes are proposed by using the Lasserre’s reconstruction algorithm and the coherent intertwiners. Such kind of geometric operators have expected semiclassical property by the definition. The consistent semiclassical limit with respect to the semiclassical D-polytopes can be obtained for the usual D-volume operator in all dimensional loop quantum gravity by fixing its undetermined regularization factor case by case.

The Volume of the Quiver Vortex Moduli Space

We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

A New Air Traffic Flow Management User-Driven Prioritisation Process for Low Volume Operator in Constraint: Simulations and Results

This document presents a new potential feature for the User Driven Prioritisation Process (UDPP) concept to give access and flexibility to Airspace Users (AUs) when they operate a low number of flights involved in a particular hotspot, a.k.a., Low Volume Users in Constraint (LVUC). Capacity constraints and congestion in the Air Traffic Management system impose delay to flights that cause large costs on airlines and passengers alike, with no significant capacity increases expected in the near-nor medium-term. Current UDPP features such as Enhanced Slot Swapping can increase flexibility for AUs to adapt their operations during capacity constrained situations. However, AUs are often impacted in their flight schedules by constraints that only affect a reduced number of flights, thus being in a situation of reduced flexibility—or no flexibility at all—to prioritise those flights. Some AUs are more vulnerable to this problem because they typically operate a low number of flights, e.g., business aviation. The new method proposed, named Flexible Credits for LVUC (FCL), is based on the use of “credits”, as a virtual currency, to increase the flexibility of LVUCs irrespective of the number of flights operated or affected by delay. FCL aims at facilitating the smooth coordination between AUs during the optimisation of their operations across multiple constraints and over the time. An initial set of simulations performed under credible conditions are presented to preliminarily analyse the feasibility and limitations of the method and to shed light on future research aspects. A first empirical evidence is given in this paper showing that increasing flexibility for LVUCs is possible without jeopardising equity.