description of projections in the convex hull of two surjective linear isometries (carrying a weighted composition operator form) on absolutely continuous function space AC(X, E), where X is a compact subset of \\(\\mathbb R\\) with at least two points and E is a strictly convex normed ...
absolutely continuous function absolutely continuous measure Absolutely convex envelope absolute-value computer absolution absolutism absorb absorbance absorbancy absorbed charge absorbed dose absorbed moisture absorbed-dose rate absorbency absorbency index
(2.18) It is well known that any convex function is locally absolutely continuous (see, e.g., [59] Proposition 17 of Chapter 5) that is, x2 f (x2) − f (x1) = f (u−)du, x1 0 < x1 < x2 < . (2.19) As the lefthand derivative f (x−) of the convex function...
Lemma . ([, Lemma .] or [, Lemma .]) Let f : [a, b] ⊆ R → R be an n-time differentiable function such that f (n–)(x) for n ∈ N is absolutely continuous on [a, b]. Then the identity b f (x) dx = n– (b – t)...
14.On level sets of E-convex function and E-quasiconvex function有关E-凸函数和E-拟凸函数的水平集 15.A convex lens is used to concentrate rays of light.凸透镜用于聚集光线。 16.On the Existence of Minimal Convex Generated Set in the Open Set of E~n;E~n中的开集不存在最小凸生成集 ...
By Theorem 1.18, f, a proper convex function, is necessarily continuous on ri dom f. As is seen from this theorem, a convex function is continuous in dom f and may have a point of discontinuity only in its boundary. In order to characterize the case in which there is no such ...
Proof. It is easy to see that for any locally absolutely continuous function f :(a,b)→R,we have the identity (2.2) ,x (t−a)f′ (t)dt+,b (t−b)f′ (t)dt=f(x)−,b f(t)dt,for any x∈(a,b)where f′ is the derivative of f which exists a.e. on(a,b) .4 ...
convex function is bounded below. We shall say that a function is strongly convex if it is universally measurable, bounded below, and if for all ×∈ K and all (Radon) laws μ on K with barycentre× (29) we have (50.1)f(x)≤∫kf(y)μ(dy) One defines similarly strongly concave,...
Notions of a boundedly convex function and of a Lipschitz-continuous function are extended to the case of functions on pseudo-topological vector spaces. It is proved that for good pseudo-topologizers Ψ, any continuous Ψ-boundedly convex function is Ψ-differentiable and its derivative is...
Let f : Δ = [a, b]× [c, d] ⊂ [0, ∞)2→ [0, ∞) be an absolutely continuous function onΔ. IfMathMLis s-convex function on the co-ordinates onΔ, for some fixed s∈ (0, 1] and q≥ 1, then one has the inequality: MathML for some fixed r1, r2∈ [0, 1]. ...