Schur-convex function Schur-convex function In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function formula_1 that for all formula_2 such that formula_3 is majorized by formula_4, one has that formula_5. Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments). A function "f" is 'Schur-concave' if its negative, -"f", is Schur-convex. If "f" is symmetric and all first partial derivatives exist, then "f" is Schur-convex
Focal cloud antennas might benefit from minor deformation, producing a footprint encompassing satellite tidal motion. Focal cloud A focal cloud is the collection of focal points of an "imperfect" lens or parabolic reflector whether optical, electrostatic or electromagnetic. This includes parabolic antennas and lens-type reflective antennas of all kinds. The effect is analogous to the circle of confusion in photography. In a perfect lens or parabolic reflector, rays parallel to the device's axis striking the lens or reflector all pass through a single point, the focal point. In an imperfectly constructed lens or reflector, rays passing through different parts of the element