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Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points. 

Abstract The merging of photocatalysis with halogen‐atom transfer (XAT) processes has proven to be a versatile tool for the generation of carbon‐centered radicals in organic synthesis. XAT processes are unique in that they generate radicals without requiring the use of strong reductants necessary for the traditional single electron transfer (SET) activation of halides. Pathways to achieve XAT in synthetic applications can be categorized into three major sections: i) heteroatom‐based activators, ii) metal‐based activators, and iii) carbon‐based activators among which α‐aminoalkyl radicals have taken the center stage. Access to these α‐aminoalkyl radicals as XAT reagents has gained significant attention in the past few years due to the robustness of the reactions, the simplicity of the reagents required, and the broadness of their applications. Generation of these α‐aminoalkyl radicals is simply achieved through the single electron oxidation of tertiary amines, which after deprotonation at the α‐position generates the α‐aminoalkyl radicals. Due to the wide scope of tertiary amines available and the tunable nucleophilicity of α‐aminoalkyl radical formed, this strategy has become an attractive alternative to heteroatom/metal‐based radicals for XAT. In this minireview, we focus our attention on recent (2020–2023) developments and uses of this robust technology to mediate XAT processes. 

Warming-induced carbon loss through terrestrial ecosystem respiration ( Re ) is likely getting stronger in high latitudes and cold regions because of the more rapid warming and higher temperature sensitivity of Re ( Q 10 ). However, it is not known whether the spatial relationship between Q 10 and temperature also holds temporally under a future warmer climate. Here, we analyzed apparent Q 10 values derived from multiyear observations at 74 FLUXNET sites spanning diverse climates and biomes. We found warming-induced decline in Q 10 is stronger at colder regions than other locations, which is consistent with a meta-analysis of 54 field warming experiments across the globe. We predict future warming will shrink the global variability of Q 10 values to an average of 1.44 across the globe under a high emission trajectory (RCP 8.5) by the end of the century. Therefore, warming-induced carbon loss may be less than previously assumed because of Q 10 homogenization in a warming world. 

A two-dimensional (2D) topological insulator exhibits the quantum spin Hall (QSH) effect, in which topologically protected conducting channels exist at the sample edges. Experimental signatures of the QSH effect have recently been reported in an atomically thin material, monolayer WTe 2 . Here, we directly image the local conductivity of monolayer WTe 2 using microwave impedance microscopy, establishing beyond doubt that conduction is indeed strongly localized to the physical edges at temperatures up to 77 K and above. The edge conductivity shows no gap as a function of gate voltage, and is suppressed by magnetic field as expected. We observe additional conducting features which can be explained by edge states following boundaries between topologically trivial and nontrivial regions. These observations will be critical for interpreting and improving the properties of devices incorporating WTe 2 . Meanwhile, they reveal the robustness of the QSH channels and the potential to engineer them in the monolayer material platform.