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   Chen, Teng-Teng; Li, Wan-Lu; Li, Jun; Wang, Lai-Sheng (February 2019, Chemical Science)

   Despite the importance of bulk lanthanide borides, nanoclusters of lanthanide and boron have rarely been investigated. Here we show that lanthanide–boron binary clusters, La 2 B x − , can form a new class of inverse-sandwich complexes, [Ln(η x -B x )Ln] − ( x = 7–9). Joint experimental and theoretical studies reveal that the monocyclic B x rings in the inverse sandwiches display similar bonding, consisting of three delocalized σ and three delocalized π bonds. Such monocyclic boron rings do not exist for bare boron clusters, but they are stabilized by the sandwiching lanthanide atoms. An electron counting rule is proposed to predict the sizes of the B x ring that can form stable inverse sandwiches. A unique (d-p)δ bond is found to play important roles in the stability of all three inverse-sandwich complexes. 

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5. The factorisation property of l ^∞ ( X_k )

   https://doi.org/10.1017/s0305004120000304  

   LECHNER, RICHARD; MOTAKIS, PAVLOS; MÜLLER, PAUL F.X.; SCHLUMPRECHT, THOMAS (September 2021, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In this paper we consider the following problem: let X k , be a Banach space with a normalised basis ( e (k, j) ) j , whose biorthogonals are denoted by $${(e_{(k,j)}^*)_j}$$ , for $$k\in\N$$ , let $$Z=\ell^\infty(X_k:k\kin\N)$$ be their l ∞ -sum, and let $$T:Z\to Z$$ be a bounded linear operator with a large diagonal, i.e. , $$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$ Under which condition does the identity on Z factor through T ? The purpose of this paper is to formulate general conditions for which the answer is positive. 

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