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We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Holder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.
Reproducibility is a defining feature of science, but the extent to which it characterizes current research is unknown. We conducted replications of 100 experimental and correlational studies published in three psychology journals using high-powered designs and original materials when available. Replication effects were half the magnitude of original effects, representing a substantial decline. Ninety-seven percent of original studies had statistically significant results. Thirty-six percent of replications had statistically significant results; 47% of original effect sizes were in the 95% confidence interval of the replication effect size; 39% of effects were subjectively rated to have replicated the original result; and if no bias in original results is assumed, combining original and replication results left 68% with statistically significant effects. Correlational tests suggest that replication success was better predicted by the strength of original evidence than by characteristics of the original and replication teams.
Food system innovations will be instrumental to achieving multiple Sustainable Development Goals (SDGs). However, major innovation breakthroughs can trigger profound and disruptive changes, leading to simultaneous and interlinked reconfigurations of multiple parts of the global food system. The emergence of new technologies or social solutions, therefore, have very different impact profiles, with favourable consequences for some SDGs and unintended adverse side-effects for others. Stand-alone innovations seldom achieve positive outcomes over multiple sustainability dimensions. Instead, they should be embedded as part of systemic changes that facilitate the implementation of the SDGs. Emerging trade-offs need to be intentionally addressed to achieve true sustainability, particularly those involving social aspects like inequality in its many forms, social justice, and strong institutions, which remain challenging. Trade-offs with undesirable consequences are manageable through the development of well planned transition pathways, careful monitoring of key indicators, and through the implementation of transparent science targets at the local level.