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Schur-concave
In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function (mathematics), function f: \mathbb^d\rightarrow \mathbb that for all x,y\in \mathbb^d such that x is majorization, majorized by y, one has that f(x)\le f(y). Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is Convex function, convex and Symmetric function, symmetric is also Schur-convex. The opposite Strict conditional, implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments). Schur-concave function A function ''f'' is 'Schur-concave' if its negative, -''f'', is Schur-convex. Schur-Ostrowski criterion If ''f'' is symmetric and all first partial derivatives exist, then ''f'' is Schur-convex if and only if (x_i - x_j)\left(\frac - \frac\right) \ge 0 for all x \in \mathbb^d holds for all 1≤''i''≠''j''≤''d''. Examples * f(x)=\min(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rényi Entropy
In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. The Rényi entropy is important in ecology and statistics as index of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of can be calculated explicitly because it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context of randomness extractors. Definition The Rényi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |