In this paper, we determine the Hausdorff dimension of set points with divergent trajectories on product certain homogeneous spaces. The flow is allowed to be weighted respect factors in space. result derived from its counterpart Diophantine approximation. doing this, introduce a notion jointly singular matrix tuples, and extend formula for matrices such tuples.

A transitive (infinite) permutation group which has m orbits on ordered pairs of distinct points has at least mn−1 orbits on ordered n-tuples. This is best possible, and groups attaining the bound can be characterised.

We use the ASTRID cosmological hydrodynamic simulation to investigate properties and evolution of triple quadruple Massive Black Hole (MBH) systems at $z = 2-3$. Only a handful MBH tuple have been detected date. In ASTRID, we find $4\%$ $M_{\rm BH}>10^7\,M_\odot$ are in tuples with $\Delta r_{\rm max} < 200\,{\rm kpc}$. The span range separations majority observable AGN r \sim 50-100$ kpc. They...