Appendix 2 - exemplary text
We shall use the following notations to indicate the asymptotic behavior of functions. The notation O(f(N)) will denote some functions of N (possibly a different function at each occurrence) whose absolute value, when divided by f(N) is ultimately bounded above by some positive constant. The notation W(f(N)) is defined analogously with "absolute" omitted and with "above" replaced by "below". Similarly o(f(N)) will denote some functions of N whose absolute value, when divided by f(N), tends to zero. The notation W(f(N)) is defined analogously, with "absolute" omitted and with zero replaced by "infinity". Roughly speaking, O(f(N)), W(f(N)), o(f(N)),and W(f(N)) denote functions that grow at most as rapidly as, at least as rapidly, less rapidly than, and more rapidly than f(N). The notation U(f(N)) will denote a factor of the form expO(f(N)). Thus U(1) denotes a function bounded between positive constants, and if f(N) = o(1), then U(f(N)) is of the form 1+O(f(N)). Finally, we shall say that f(N) is asymptotic to g(N) if their ratio tends to unity. Since all our results concern asymptotic behavior, we need not worry if any of our arguments or constructions fail for the first few values of N.
