A pre-registered short-term forecasting study of COVID-19 in Germany and Poland during the second wave.

Disease modelling has had considerable policy impact during the ongoing COVID-19 pandemic, and it is increasingly acknowledged that combining multiple models can improve the reliability of outputs. Here we report insights from ten weeks of collaborative short-term forecasting of COVID-19 in Germany and Poland (12 October-19 December 2020). The study period covers the onset of the second wave in both countries, with tightening non-pharmaceutical interventions (NPIs) and subsequently a decay (Poland) or plateau and renewed increase (Germany) in reported cases. Thirteen independent teams provided probabilistic real-time forecasts of COVID-19 cases and deaths. These were reported for lead times of one to four weeks, with evaluation focused on one- and two-week horizons, which are less affected by changing NPIs. Heterogeneity between forecasts was considerable both in terms of point predictions and forecast spread. Ensemble forecasts showed good relative performance, in particular in terms of coverage, but did not clearly dominate single-model predictions. The study was preregistered and will be followed up in future phases of the pandemic.

Scaling laws for critical manifolds in polycrystalline materials.

We study the surfaces of lowest energy through model polycrystalline materials in two and three dimensions. When the grain boundaries are sufficiently weak, these critical manifolds (CM's) lie entirely on grain boundaries, while when the grain boundaries are strong, cleavage occurs. A scaling theory for the intergranular to transgranular transition of CM's is developed. The key parameters are the average grain size g, the ratio of grain boundary to the grain interior energy, epsilon, and the sample size L. The key result is that a critical length scale exists, L(c)(g,epsilon), so that on short length scales lL(c), the critical manifold is rough. We develop a scaling theory for L(c) and find that in two dimensions L(c) approximately gx(y(2)), while in three dimensions L(c) approximately g exp(bx(y(3))), where x=epsilon/(1-epsilon) and b is a constant. Data from realistic polycrystalline grain structures are used to test the scaling theory. The exact lowest energy surface through model grain structures is found using a mapping to the minimum-cut/maximum-flow problem in computer science. As a function of grain-boundary energy, we observe the crossover from grain-boundary rupture to mixed mode failure (a mixture of transgramular and intergranular modes) and finally cleavage and that the two-dimensional data are consistent with y(2) approximately 3.0+/-0.3, while the three-dimensional data are more difficult to analyze, but are consistent with y(3) approximately 3.5+/-1.0.

Ground state nonuniversality in the random-field Ising model.

Two attractive and often used ideas, namely, universality and the concept of a zero-temperature fixed point, are violated in the infinite-range random-field Ising model. In the ground state we show that the exponents can depend continuously on the disorder and so are nonuniversal. However, we also show that at finite temperature the thermal order-parameter exponent 1/2 is restored so that temperature is a relevant variable. Broader implications of these results are discussed.