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+%This commands provides the text for the definitions on the first
+%page.
+%This command has one parameter:
+% 1) The width of the text
+\newcommand\TOneDef[1]{%
+ \def\LineOfArray##1##2{%
+ {\ensuremath{##1}}&%
+ \begin{DisplayFormulae}{0}{0pt}{2ex plus .5ex minus .5ex}%
+ {\BigChar}{\StyleWithoutNumber}%
+ {\raggedright##2\par\vspace{\TOneInterlineDef}}%
+ \end{DisplayFormulae}\\
+ }
+ \begingroup
+ \def\LongestExpr{f(n) = \Omega(g(n))}%
+ \settowidth{\TmpLengthA}{$\LongestExpr$}
+ \parbox[t]{#1}{%
+ \setlength{\HSpace}{#1-\TmpLengthA}%
+ \begin{tabular}{@{}l@{\hspace{.2em}}|%
+ p{\HSpace}}
+ %Line 1
+ \LineOfArray{f(n) = O(g(n))}%
+ {iff \Fm[true]{\exists} positive \Fm[true]{c, n_0} such that
+ \Fm[true]{0 \leq f(n) \leq cg(n)}
+ \Fm[true]{\MathRemark[\relax]{\forall n \geq n_0}.}}
+ %Line 2
+ \LineOfArray{\LongestExpr}%
+ {iff $\exists$ positive \Fm[true]{c, n_0} such that
+ \Fm[true]{f(n) \geq cg(n) \geq 0}
+ \Fm[true]{\MathRemark[\relax]{\forall n \geq n_0}}.}
+ %Line 3
+ \LineOfArray{f(n) = \Theta(g(n))}%
+ {iff \Fm[true]{f(n) = O(g(n))} and \Fm[true]{f(n) = \Omega(g(n))}. }
+ %Line 4
+ \LineOfArray{f(n) = o(g(n))}%
+ {iff \Fm[true]{\lim_{n \to \infty} f(n)/g(n) = 0}.}
+ %Line 5
+ \LineOfArray{\lim_{n \to \infty} a_n = a}%
+ {iff \Fm[true]{\forall \epsilon > 0}, \Fm[true]{\exists n_0} such that
+ \Fm[true]{\vert a_n - a\vert < \epsilon\MathRemark{\forall n \geq n_0}}.}
+ %Line 6
+ \LineOfArray{\sup S}%
+ {least \Fm[true]{b \in \Real} such that \Fm[true]{b \geq s\MathRemark{\forall s \in S}}.}
+ %Line 7
+ \LineOfArray{\inf S}%
+ {greatest \Fm[true]{b \in \Real} such that \Fm[true]{b \leq s\MathRemark{\forall s \in S}}.}
+ %Line 8
+ \LineOfArray{\liminf_{n \to \infty} a_n}%
+ {\def\FirstPart{\lim_{n\to\infty} \inf \{\mbox{}}%
+ \Fm[true]{\FirstPart a_i \mid i \geq n,}%
+ \FmPartB{\FirstPart}{\MathRemark[\relax]{i \in \Natural}}\}
+ }
+ %Line 9
+ \LineOfArray{\limsup_{n \to \infty} a_n}%
+ {\def\FirstPart{\lim_{n\to\infty} \sup \{\mbox{}}%
+ \Fm[true]{\FirstPart a_i \mid i\geq n,}%
+ \FmPartB{\FirstPart}{\MathRemark[\relax]{i \in \Natural}}\}
+ }
+ %Line 10
+ \LineOfArray{\binom{n}{k}}{Combinations: Size $k$ subsets of a size $n$ set.}
+ %Line 11
+ \LineOfArray{\cycle{n}{k}}%
+ {Stirling numbers (1\textsuperscript{st} kind):
+ Arrangements of an $n$ element set into $k$ cycles.}
+ %Line 12
+ \LineOfArray{\SousEnsemble{n}{k}}%
+ {Stirling numbers (2\textsuperscript{nd} kind):
+ Partitions of an $n$ element set into $k$ non-empty sets.}
+ %Line 13
+ \LineOfArray{\eul{n}{k}}%
+ {1\textsuperscript{st} order Eulerian numbers:
+ Permutations \Fm[true]{\pi_1\pi_2\ldots\pi_n} on \Fm[true]{\{1,2, \ldots, n\}} with $k$ ascents.}
+ %Line 14
+ \LineOfArray{\euls{n}{k}}{2\textsuperscript{nd} order \mbox{Eulerian} numbers.}
+ %Line 15
+ \LineOfArray{C_n}{Catalan Numbers: Binary trees with \Fm[true]{n + 1} vertices.}
+ \end{tabular}
+ }%
+ \endgroup
+}