Evaluating Quasilocal Energy and Solving Optimal Embedding Equation at Null Infinity

We study the limit of quasilocal energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi–Sachs energy-momentum at a retarded time. The quasilocal mass in Wang and Yau (Phys Rev Lett 102(2):021101, 2009; Commun Math Phys 288(3):919–942, 2009) is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.     

Harnack inequalities for graphs with non-negative Ricci curvature

We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.     

Rhodamine‐Ethylenediol,A Novel Vital Fluorescent Probe for Labelling Alkaline Phosphatase‐Rich Organelles

Zebrafish have received considerable attention as an organism‐based model in the development of pharmacological agents.^1,2 Many small molecules applied to zebrafish show important behaviours and may constitute new kinds of markers for clinical purposes.³ Analysis of these molecules can facilitate the development of useful tools for monitoring environmental changes.⁴ Many chemicals that are toxic to the environment are known to influence the sensory systems of humans⁵ and fish.⁶ One important sensory system in all fish is the lateral line organ,⁷ which is readily accessible for the assessment of environmental changes.⁸ Neuromasts, which are located on the surface of the fish body, are one of the major components of the lateral lines of the zebrafish.⁹ Copper‐enriched water is known to affect the olfactory system in fish. Therefore, small molecules that induce specific patterns in the neuromasts of zebrafish should provide an important animal model with which to explore the effects of environmental changes on the sensory system.^10,11 Recently, chemical sensors based on the rhodamine skeleton¹² have been designed to specifically detect metal ions, such as Cu(II)¹³ and Fe(III)/Hg(II),¹⁴ in zebrafish. However, there has been no report of these rhodamine derivatives used in the specific recognition of the sensory system of zebrafish. Commonly, the sensory system is studied with antibody staining assays of scarified fish. Here, we report that a new rhodamine derivative can be used as a fluorescent chemical probe to visualize the neuromasts and intestinal villi of living zebrafish. Based on the specific recognition of this area in zebrafish, we narrowed the possible enzymes targeted by this rhodamine probe to alkaline phosphatase and confirmed this with a binding assay. It is a well‐recognized challenge to develop a fluorescent chemical probe that specifically recognizes a particular enzyme. Furthermore, the transfer of phosphate groups to certain enzymes can activate their catalytic reactivity, triggering a cascade reaction in a signal transduction pathway. The alkaline‐phosphatase‐specific recognition by this rhodamine derivative may be applicable to clinical purposes.     

Iron-catalyzed selective reduction of nitro compounds to amines

An efficient reduction of the nitro group with a catalytic amount of Fe(acac)₃ and TMDS in THF at 60 °C affording the corresponding amine is described.