3-2 Relative asymptotic growths
Indicate for each pair of expressions \((A, B)\) in the table below, whether \(A\) is \(O\), \(o\), \(\Omega\), \(\omega\), or \(\Theta\) of \(B\). Assume that \(k \ge 1\), \(\epsilon > 0\), and \(c > 1\) are constants. Your answer should be in the form of the table with "yes" or "no" written in each box.
\[
\begin{array}{ccccccc}
A & B & O & o & \Omega & \omega & \Theta \\\\
\hline
\lg^k n & n^\epsilon & yes & yes & no & no & no \\\\
n^k & c^n & yes & yes & no & no & no \\\\
\sqrt n & n^{\sin n} & no & no & no & no & no \\\\
2^n & 2^{n / 2} & no & no & yes & yes & no \\\\
n^{\lg c} & c^{\lg n} & yes & no & yes & no & yes \\\\
\lg(n!) & \lg(n^n) & yes & no & yes & no & yes
\end{array}
\]
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