- #71

evinda

Gold Member

MHB

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Then we check how many times the while-loop is executed. There is the following lemma.

View attachment 7628

First of all, how do we get that $\Omega\left(\frac{x}{\log{x}}\right)=\Omega((\log{n})^3 (\log{\log{n}})^2)$ ?

I got that $\frac{x}{\log{x}}=\frac{8 \lceil \log{n}\rceil^3 (\log{\log{n}})^3}{\log{8}+3 \log{(\lceil \log{n}\rceil)+3 \log{(\log{\log{n}})}}}$... But is this in $\Omega((\log{n})^3 (\log{\log{n}})^2)$ ?

Also how do we get that $n^{\frac{x^{\frac{2}{3}}}{3}} <\Pi<n^{x^{\frac{2}{3}}}$ ? (Thinking)

View attachment 7628

First of all, how do we get that $\Omega\left(\frac{x}{\log{x}}\right)=\Omega((\log{n})^3 (\log{\log{n}})^2)$ ?

I got that $\frac{x}{\log{x}}=\frac{8 \lceil \log{n}\rceil^3 (\log{\log{n}})^3}{\log{8}+3 \log{(\lceil \log{n}\rceil)+3 \log{(\log{\log{n}})}}}$... But is this in $\Omega((\log{n})^3 (\log{\log{n}})^2)$ ?

Also how do we get that $n^{\frac{x^{\frac{2}{3}}}{3}} <\Pi<n^{x^{\frac{2}{3}}}$ ? (Thinking)