Before revision:  the second is data structure for answering range search queries with semi-algebraic ranges in  O(\log n) time ; the third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in \mathbb{R}^{d} in O(\log^2 n) time ;
After revision:  the second is data structure for answering range search queries with semi-algebraic ranges in O(\log n) time , with O(n^{t+\epsilon third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in \mathbb{R}^{d} in O(\log^2 n) time ;