MHB Master Theorem-Relation of the two functions

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Functions Master
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

I want to solve the recurrence relation

$$T(n)=4T{\left( \frac{n}{3} \right)}+n \log{n}.$$

I thought to use the Master Theorem.

We have $a=4, b=3, f(n)=n \log{n}$.

$\log_b{a}=\log_3{4}$

$n^{\log_b{a}}=n^{\log_3{4}}$

How can we find a relation between $n^{\log_{3}{4}}$ and $n \log{n}$ ? (Thinking)
 
Physics news on Phys.org
Hey evinda!

A polynomial dominates the logarithm doesn't it? 🤔
More specifically, for any $\varepsilon>0$ we have that $\ln n < n^\varepsilon$ for sufficiently large $n$.
Proof by L'Hôpital's rule:
$$\lim_{n\to\infty} \frac{\ln n}{n^\varepsilon} = \lim_{n\to\infty} \frac{\frac 1n}{\varepsilon n^{\varepsilon-1}} = \lim_{n\to\infty} \frac{1}{\varepsilon n^\varepsilon} = 0$$
🤔
 
Klaas van Aarsen said:
Hey evinda!

A polynomial dominates the logarithm doesn't it? 🤔
More specifically, for any $\varepsilon>0$ we have that $\ln n < n^\varepsilon$ for sufficiently large $n$.
Proof by L'Hôpital's rule:
$$\lim_{n\to\infty} \frac{\ln n}{n^\varepsilon} = \lim_{n\to\infty} \frac{\frac 1n}{\varepsilon n^{\varepsilon-1}} = \lim_{n\to\infty} \frac{1}{\varepsilon n^\varepsilon} = 0$$
🤔

I see... Thanks a lot! (Blush)
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Back
Top