Proper convex function Proper convex function In mathematical analysis (in particular convex analysis) and optimization, a proper convex function is a convex function "f" taking values in the extended real number line such that for at least one "x" and for every "x". That is, a convex function is "proper" if its effective domain is nonempty and it never attains formula_3. Convex functions that are not proper are called "improper convex functions". A "proper concave function" is any function "g" such that formula_4 is a proper convex function. For every proper convex function "f" on R there exist some "b" in R and
Anticrepuscular rays Anticrepuscular rays Anticrepuscular rays, or antisolar rays, are atmospheric optical phenomena similar to crepuscular rays, but appear opposite of the Sun in the sky. Anticrepuscular rays are nearly parallel, but appear to converge toward the antisolar point due to linear perspective. Anticrepuscular rays are most frequently visible around sunrise or sunset. Appearing to radiate from the Sun, crepuscular rays usually look much brighter than anticrepuscular rays. This is because the atmospheric light scattering making the crepuscular rays visible occurs at low angles to the horizon (see Mie theory). Although anticrepuscular rays appear to converge toward the antisolar point, the convergence