χ-Octamolybdate [Mo(V)4Mo(VI)4O24](4-): an unusual small polyoxometalate in partially reduced form from nonaqueous solvent reduction

The first mixed-valent octamolybdate anion [Mo(V)(4)Mo(VI)(4)O(24)](4-), termed χ-octamolybdate due to its shape, was obtained by partial reduction of α-(Bu(4)N)(4)-[Mo(8)O(26)] in dry acetonitrile. Single-crystal X-ray structure analysis revealed that the anion includes an unusual Mo(V)(4)O(8) cubane-like cluster core, whose four side faces are capped by four MoO(4) units to form a crosslike cluster. X-ray photoelectron spectroscopy (XPS) and bond valence sum (BVS) calculations were carried out to validate the presence of mixed-valent Mo centers.     

On a family of matrix equalities that involve multiple products of generalized inverses

Aequationes mathematicae - One of the fundamental matrix equalities that involve multiple products of matrices and their generalized inverses is given by $$A_1B_1^{-}A_2B_2^{-} \ldots...