Boron–Sulfur and Boron–Selenium Compounds—From Unique Molecular Structural Principles to Novel Polymeric Materials

An immensely broad scope of structural motifs is covered by the intriguing chemistry of boron–sulfur and boron–selenium compounds. Characteristic structural features of the binary boron sulfides can be applied to understand and rationalize the three-dimensional atomic arrangement in macromolecular anionic frameworks such as that of Li_6+2x[B₁₀S₁₈]S_x (x≈2, shown in the picture; ○: S, •: B, ○: Li). Of interest are also the ion dynamics of lithium thioborates and a persubstituted B₁₂ icosahedron that was prepared by high-temperature synthesis and features a characteristic planar B₃Se₂ ring at the chelating ligand.     

Indecomposable representations for real roots of a wild quiver

For a given quiver and dimension vector, Kac has shown that there is exactly one indecomposable representation up to isomorphism if and only if this dimension vector is a positive real root. However, it is not clear how to compute these indecomposable representations in an explicit and minimal way, and the properties of these representations are mostly unknown. In this note we study representations of a particular wild quiver. We define operations which act on representations of this quiver, and using these operations we construct indecomposable representations for positive real roots, compute their endomorphism rings and show that these representations are tree representations. The operations correspond to the fundamental reflections in the Weyl group of the quiver. Our results are independent of the characteristic of the field.     

Projection estimates for harmonic measure

Stochastic proofs of the Beurling projection theorem and the Hall projection theorem for harmonic measure are given. Somed-dimensional versions (for alld>1) which follow from this