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
Compound refractive lens Compound refractive lens A Compound refractive lens (CRL) is a series of individual lenses arranged in a linear array in order to achieve focusing of X-rays in the energy range of 5-40 keV. They are an alternative to the KB mirror. For all materials the refractive index for X-rays is close to 1, hence a single conventional lens for X-rays has an extremely long focal length (for practical lens sizes). In addition, X-rays attenuate as they pass through a material so that conventional lenses for X-rays have long been considered impractical. The CRL gets its reasonably short focal length, on