(a,b)-trees: Fix a couple of typos

03-abtree/abtree.tex

@@ -105,7 +105,7 @@ Therefore in each tree, we have $n \le b^h-1$, so $h \ge \log_b (n+1)$.
 \subsection{Searching for a~key}
 
 $\alg{Find}(x)$ follows the general algorithm for multi-way trees.
-It visits $\O(\log_a n) = \O(\log n/\log a)$ nodes. In each node, it compares~$x$ with
+It visits $\O(\log_a n)$ nodes, which is $\O(\log n/\log a)$. In each node, it compares~$x$ with
 all keys of the node, which can be performed in time $\O(\log b)$ by binary search.
 In total, we spend time $\Theta(\log n \cdot \log b / \log a)$.
 
@@ -175,7 +175,7 @@ sibling is large, we can fix our problem by borrowing a~key from it.
 Let us be exact. Suppose that we have an undersized node~$v$ with
 $a-2$ keys and this node has a~left sibling~$\ell$ separated by a~key~$p$ in their
 common parent. If there is no left sibling, we use the right sibling and follow
-A~MIRROR image of the procedure.
+a~mirror image of the procedure.
 
 If the sibling has only~$a$ children, we merge nodes~$v$ and~$\ell$ to a~single
 node and we also move the key~$p$ from the parent there. This creates a~node with
@@ -290,7 +290,7 @@ performs $\O(m)$ node modifications.}
 \proof
 We define the cost of an~operation as the number of nodes it modifies.
 We will show that there exists a~potential~$\Phi$ such that the amortized cost
-of splitting and merging with respect to~$\Phi$ is at most~0 and the amortized
+of splitting and merging with respect to~$\Phi$ is zero or negative and the amortized
 cost of the rest of \alg{Insert} and \alg{Delete} is constant.
 
 The potential will be a~sum of node contributions. Every node with $k$~keys