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In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. This means that for any minimization problem, called the primal problem, the solution to the primal problem is always greater than or equal to the solution to the dual maximization problem.
The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.
Sri Sultan Hamengkubuwono IX (Hanacaraka: ꦯꦿꦶꦯꦸꦭ꧀ꦡꦟ꧀ꦲꦩꦼꦁꦑꦸꦨꦸꦮꦟ꧇꧙꧇; 12 April 1912 – 2 October 1988 [a]), often abbreviated as HB IX, was an Indonesian politician and Javanese royal who was the second vice president of Indonesia, the ninth sultan of Yogyakarta, and the first governor of the Special Region of Yogyakarta.