On melting of boron phosphide under pressure

Melting of cubic boron phosphide, BP has been studied at pressures to 9 GPa using synchrotron X-ray diffraction and electrical resistivity measurements. It has been found that above 2.6 GPa BP melts congruently, and the melting curve exhibits negative slope -60(7) K/GPa, which is indicative of a higher density of the melt as compared to the solid phase.Comment: 3 pages, 1 figur

Generically multiple transitive algebraic group actions

With every nontrivial connected algebraic group $G$ we associate a positive integer ${\rm gtd}(G)$ called the generic transitivity degree of $G$ and equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$ admits an open orbit. We show that ${\rm gtd}(G)\leqslant 2$ (respectively, ${\rm gtd}(G)=\nobreak 1$) for all solvable (respectively, nilpotent) $G$, and we calculate ${\rm gtd}(G)$ for all reductive $G$. We prove that if $G$ is nonabelian reductive, then the above maximal $n$ is attained for $X=G/P$ where $P$ is a proper maximal parabolic subgroup of $G$ (but not only for such homogeneous spaces of $G$). For every reductive $G$ and its proper maximal parabolic subgroup $P$, we find the maximal $r$ such that the diagonal action of $G$ (respectively, a Levi subgroup $L$ of ~$P$) on $(G/P)^r$ admits an open $G$-orbit (respectively, $L$-orbit). As an application, we obtain upper bounds for the multiplicities of trivial components in some tensor product decompositions. As another application, we classify all the pairs $(G, P)$ such that the action of $G$ on $(G/P)^3$ admits an open orbit, answering a question of {\sc M. Burger}.Comment: Revised version, 30 page