Corrected bugs reported by Koubek.

[saga.git] / notation.tex

diff --git a/notation.tex b/notation.tex
index cc44d4a744668e56950f094eabf2695fb82d7f11..e02cccf67e49f23f78e77d15a54e569d54adbcb5 100644 (file)
--- a/notation.tex
+++ b/notation.tex
@@ -113,7 +113,7 @@ better define it carefully, following \cite{diestel:gt}:
 
 \defn A~\df{multigraph} is an ordered triple $(V,E,M)$, where $V$~is the
 set of vertices, $E$~is the set of edges, taken as abstract objects disjoint
-with the vertices, and $M$ is a mapping $E\rightarrow V \cup {V \choose 2}$
+with the vertices, and $M$ is a mapping $E\rightarrow {V \choose 2} \cup {V \choose 1}$
 which assigns to each edge either a pair of vertices or a single vertex
 (if the edge is a loop).
 
@@ -180,7 +180,7 @@ To quote Pettie \cite{pettie:onlineverify}: ``In the field of algorithms \& comp
 Ackermann's function is rarely defined the same way twice. We would not presume to buck
 such a~well-established precedent. Here is a~slight variant.''}
 We will use the definition by double recursion given by Tarjan \cite{tarjan:setunion},
-which is predominant in the literature on graph algorithms:
+which is predominant in the literature on graph algorithms.
 
 \defn\id{ackerdef}%
 The \df{Ackermann's function} $A(x,y)$ is a~function on non-negative integers defined as follows:
@@ -207,8 +207,10 @@ A(4,3) &= A(3,A(4,2)) = A(3,4) = A(2,A(3,3)) = A(2,A(2,A(3,2))) = \cr
        &= A(2,A(2,4)) = 2\tower(2\tower 4) = 2\tower 65536. \cr
 }$$
 
-\para
-Three functions related to the inverse of the function~$A$ are usually considered:
+\paran{Inverses}%
+As common for functions of multiple parameters, there is no single function which
+could claim the title of the only true Inverse Ackermann's function.
+The following three functions related to the inverse of the function~$A$ are often considered:
 
 \defn\id{ackerinv}%
 The \df{$i$-th row inverse} $\lambda_i(n)$ of the Ackermann's function is defined by: