MHB Solving Recurrence Relations Using Master Theorem/Akra-Bazzi

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The Master Theorem is a method for analyzing the time complexity of divide-and-conquer algorithms, providing a straightforward way to solve recurrence relations of the form T(n) = aT(n/b) + f(n). For the first recurrence, T(n) = 3T([n/3]) + n, it can be solved using the Master Theorem, yielding T(n) = Θ(n). The second recurrence, T(n) = T([n/4]) + T([n/3]) + n, does not fit the Master Theorem's criteria and should be approached with the Akra-Bazzi method. The third recurrence, T(n) = 2T([n/4]) + √n, can also be solved using the Master Theorem, resulting in T(n) = Θ(n^0.5). Understanding these theorems is crucial for efficiently solving recurrence relations in algorithm analysis.
vapatel
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first state whether it can be solved using the Master Theorem, and if it can then use that. Otherwise, use the Akra-Bazzi formula.

1. T(n) = 3T([n/3])+n

2. T(n) = T([n/4])+T([n/3])+n

3. T(n) = 2T([n/4])+√n
 
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Okay, what is the "master theorem" and what is the "Akra-Bazzi theorem"?
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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