$a_i$-invariants of powers of ideals.

Inspired by the recent work of Lu and O'Rourke, we study the $a_i$-invariants of (symbolic) powers of some graded ideals. The first scenario is when $I$ and $J$ are two graded ideals in two distinct polynomial rings $R$ and $S$ over a common field $\mathbb{K}$. We study the $a_i$-invariants of the powers of the fiber product via the corresponding knowledge of $I$ and $J$. The second scenario is when $I_{\Delta}$ is the Stanley-Reisner ideal of a $k$-dimensional simplicial complex $\Delta$ with $k\ge 2$. We investigate the $a_i$-invariants of the symbolic powers of $I_{\Delta}$.