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Abstract A version of the singular Yamabe problem in smooth domains in a closed manifoldyields complete conformal metrics with negative constant scalar curvatures.In this paper, we study the blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to a certain limit set of a lower dimension.We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold.In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere.In certain cases, the blow-up set can be the entire manifold.We will demonstrate by examples that these results are optimal. 

To address the rapid growth of scientific publications and data in biomedical research, knowledge graphs (KGs) have become a critical tool for integrating large volumes of heterogeneous data to enable efficient information retrieval and automated knowledge discovery. However, transforming unstructured scientific literature into KGs remains a significant challenge, with previous methods unable to achieve human-level accuracy. Here we used an information extraction pipeline that won first place in the LitCoin Natural Language Processing Challenge (2022) to construct a large-scale KG named iKraph using all PubMed abstracts. The extracted information matches human expert annotations and significantly exceeds the content of manually curated public databases. To enhance the KG’s comprehensiveness, we integrated relation data from 40 public databases and relation information inferred from high-throughput genomics data. This KG facilitates rigorous performance evaluation of automated knowledge discovery, which was infeasible in previous studies. We designed an interpretable, probabilistic-based inference method to identify indirect causal relations and applied it to real-time COVID-19 drug repurposing from March 2020 to May 2023. Our method identified around 1,200 candidate drugs in the first 4 months, with one-third of those discovered in the first 2 months later supported by clinical trials or PubMed publications. These outcomes are very challenging to attain through alternative approaches that lack a thorough understanding of the existing literature. A cloud-based platform (https://biokde.insilicom.com) was developed for academic users to access this rich structured data and associated tools. 

Type 1 copper (T1Cu) proteins play important roles in electron transfer in biology, largely due to the unique structure of the T1Cu center, which is reflected by its spectroscopic properties. Previous reports have suggested a correlation between a high ratio of electronic absorbance at ∼450 nm to that at ∼600 nm (R = A450/A600) and a large copper(II) hyperfine coupling in the z direction (Az) in electron paramagnetic resonance (EPR). However, this correlation does not have a clear physical meaning, nor does it hold for many proteins with a perturbed T1Cu center. To address this issue, a new parameter of R′ [A450/(A450 + A600)] with a better physical meaning of a fractional SCys pseudo-σ to Cu(II) charge transfer transition intensity is defined and a quadratic relationship between R′ and Az is found on the basis of a comprehensive analysis of ultraviolet–visible absorption, EPR, and structural parameters of T1Cu proteins. We are able to find good correlations between R′ and the displacement of copper from the trigonal plane defined by the His2Cys ligands and the angle between the NHis1–Cu–NHis2 plane and the SCys–Cu–axial ligand plane, providing a structural basis for the observed correlation. These findings and analyses provide a new framework for a deeper understanding of the spectroscopic and electronic properties of T1Cu proteins, which may allow better design and applications of this important class of proteins for redox and electron transfer functions.