Quantitative Graded Semantics and Spectra of Behavioural Metrics

Behavioural metrics provide a quantitative refinement of classical two-valued behavioural equivalences on systems with quantitative data, such as metric or probabilistic transition systems. In analogy to the classical linear-time/branching-time spectrum of two-valued behavioural equivalences on transition systems, behavioural metrics come in various degrees of granularity, depending on the observer's ability to interact with the system. Graded monads have been shown to provide a unifying framework for spectra of behavioural equivalences. Here, we transfer this principle to spectra of behavioural metrics, working at a coalgebraic level of generality, that is, parametrically in the system type. In the ensuing development of quantitative graded semantics, we discuss presentations of graded monads on the category of metric spaces in terms of graded quantitative equational theories. Moreover, we obtain a canonical generic notion of invariant real-valued modal logic, and provide criteria for such logics to be expressive in the sense that logical distance coincides with the respective behavioural distance. We thus recover recent expressiveness results for coalgebraic branching-time metrics and for trace distance in metric transition systems; moreover, we obtain a new expressiveness result for trace semantics of fuzzy transition systems. We also provide a number of salient negative results. In particular, we show that trace distance on probabilistic metric transition systems does not admit a characteristic real-valued modal logic at all

Demystifying the German ‘Armament Miracle’ During World War II. New Insights from the Annual Audits of German Aircraft Producers

Armament minister Albert Speer is usually credited with causing the boom in German armament production after 1941. This paper uses the annual audit reports of the Deutsche Revisions - und Treuhand AG for seven firms which together represented about 50% of the German aircraft producers. We question the received view by showing that in the German aircraft industry the crucial changes that triggered the upswing in aircraft production already occurred before World War II. The government decided in 1938 that aircraft producers had to concentrate on a few different types, and in 1937 that cost-plus contracts were replaced with fixed price contracts. What followed was not a sudden production miracle but a continuous development which was fueled first by learning-by-doing and then by the ongoing growth of the capital and labor endowment

Lattice parameters of Scx_{\boldsymbol{\mathsf{x}}}Al1−x_{\boldsymbol{\mathsf{1-x}}}N layers grown on GaN(0001) by plasma-assisted molecular beam epitaxy

An accurate knowledge of the lattice parameters of the new nitride Sc$_\textit{x}$Al$_\textit{1-x}$N is essential for understanding the elastic and piezoelectric properties of this compound as well as for the ability to engineer its strain state in heterostructures. Using high-resolution x-ray diffractometry, we determine the lattice parameters of 100-nm-thick undoped Sc$_\textit{x}$Al$_\textit{1-x}$N layers grown on GaN(0001) templates by plasma-assisted molecular beam epitaxy. The Sc content $\textit{x}$ of the layers is measured independently by both x-ray photoelectron spectroscopy and energy-dispersive x-ray spectroscopy and ranges from 0 to 0.25. The in-plane lattice parameter of the layers linearly increases with increasing $\textit{x}$, while their out-of-plane lattice parameter remains constant. Layers with $\textit{x}$ $\approx$ 0.09 are found to be lattice matched to GaN, resulting in a smooth surface and a structural perfection equivalent to that of the GaN underlayer. In addition, a two-dimensional electron gas is induced at the Sc$_\textit{x}$Al$_\textit{1-x}$N/GaN heterointerface, with the highest sheet electron density and mobility observed for lattice-matched conditions

Growth mechanisms in molecular beam epitaxy for GaN-(In,Ga)N core-shell nanowires emitting in the green spectral range

Using molecular beam epitaxy, we demonstrate the growth of (In,Ga)N shells emitting in the green spectral range around very thin (35 nm diameter) GaN core nanowires. These GaN nanowires are obtained by self-assembled growth on TiN. We present a qualitative shell growth model accounting for both the three-dimensional nature of the nanostructures as well as the directionality of the atomic fluxes. This model allows us, on the one hand, to optimise the conditions for high and homogeneous In incorporation and, on the other hand, to explain the influence of changes in the growth conditions on the sample morphology and In content. Specifically, the impact of the V/III and In/Ga flux ratios, the rotation speed and the rotation direction are investigated. Notably, with In acting as surfactant, the ternary (In,Ga)N shells are much more homogeneous in thickness along the NW length than their binary GaN counterparts