Optimization inferred elements

Proof. By property R5 listed at the end of Section II, the elements of the Pauli subspace S are traceless, from which we infer by Theorems 12 and 13 that the energy problem and the spectral optimization problem have optimal solutions. By Theorem 10 these solutions are characterized by the Euler equation PQ = 0. ... 

Hybrid and adaptive techniques that combine elements from both above methods have also been proposed. For example the required details of a fuzzy inference system can be tuned by means of examples and the usage of neural networks or by using some, usually global, optimization. 

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