Sufficiency of Favard's condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, with $U$ being a complex Banach space, the injectivity of all limit operators of $A$ already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of $A$, which, on the other hand, is often equivalent to the Fredholmness of $A$. As a consequence, for operators $A$ in the Wiener algebra, we can characterize the essential spectrum of $A$ on $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.

Reliability, sufficiency, and the decomposition of proper scores

Scoring rules are an important tool for evaluating the performance of probabilistic forecasting schemes. A scoring rule is called strictly proper if its expectation is optimal if and only if the forecast probability represents the true distribution of the target. In the binary case, strictly proper scoring rules allow for a decomposition into terms related to the resolution and the reliability of a forecast. This fact is particularly well known for the Brier Score. In this article, this result is extended to forecasts for finite-valued targets. Both resolution and reliability are shown to have a positive effect on the score. It is demonstrated that resolution and reliability are directly related to forecast attributes that are desirable on grounds independent of the notion of scores. This finding can be considered an epistemological justification of measuring forecast quality by proper scoring rules. A link is provided to the original work of DeGroot and Fienberg, extending their concepts of sufficiency and refinement. The relation to the conjectured sharpness principle of Gneiting, et al., is elucidated.

Potential of legume-based grassland-livestock systems in Europe

European grassland-based livestock production systems face the challenge of producing more meat and milk to meet increasing world demands and to achieve this using fewer resources. Legumes offer great potential for achieving these objectives. They have numerous features that can act together at different stages in the soil–plant–animal–atmosphere system, and these are most effective in mixed swards with a legume proportion of 30–50%. The resulting benefits include reduced dependence on fossil energy and industrial N-fertilizer, lower quantities of harmful emissions to the environment (greenhouse gases and nitrate), lower production costs, higher productivity and increased protein self-sufficiency. Some legume species offer opportunities for improving animal health with less medication, due to the presence of bioactive secondary metabolites. In addition, legumes may offer an adaptation option to rising atmospheric CO2 concentrations and climate change. Legumes generate these benefits at the level of the managed land-area unit and also at the level of the final product unit. However, legumes suffer from some limitations, and suggestions are made for future research to exploit more fully the opportunities that legumes can offer. In conclusion, the development of legume-based grassland–livestock systems undoubtedly constitutes one of the pillars for more sustainable and competitive ruminant production systems, and it can be expected that forage legumes will become more important in the future.