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In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan, [1] is a technique that provides an analytic expression for the Mellin transform of an analytic function. Page from Ramanujan's notebook stating his Master theorem. The result is stated as follows:
The master theorem always yields asymptotically tight bounds to recurrences from divide and conquer algorithms that partition an input into smaller subproblems of equal sizes, solve the subproblems recursively, and then combine the subproblem solutions to give a solution to the original problem. The time for such an algorithm can be expressed ...
Mason–Stothers theorem (polynomials) Master theorem (analysis of algorithms) (recurrence relations, asymptotic analysis) Maschke's theorem (group representations) Matiyasevich's theorem (mathematical logic) Max flow min cut theorem (graph theory) Max Noether's theorem (algebraic geometry) Maximal ergodic theorem (ergodic theory)
Master theorem (analysis of algorithms), analyzing the asymptotic behavior of divide-and-conquer algorithms; Ramanujan's master theorem, providing an analytic expression for the Mellin transform of an analytic function; MacMahon master theorem (MMT), in enumerative combinatorics and linear algebra; Glasser's master theorem in integral calculus
The cutoff function must be normalized to f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows.
Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort , the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as T ( 1 ) = 0 {\displaystyle T(1)=0} and
Percy MacMahon was born in Malta to a British military family. His father was a colonel at the time, retired in the rank of the brigadier. [1] MacMahon attended the Proprietary School in Cheltenham. At the age of 14 he won a Junior Scholarship to Cheltenham College, which he attended as a day boy from 10 February 1868 until December 1870.
A quantum master equation is a generalization of the idea of a master equation. Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a density matrix ), quantum master equations are differential equations for the entire density matrix, including off-diagonal elements.