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 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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