Let Ω denote an algebra of sets and μ a σ-finite measure. We then prove that the completion of Ω under the pseudometric d(A,B) = μ∗(A△B) is σ-algebra isomorphic and isometric to the Caratheodory Extension of Ω under the equivalence relation ∼.

We present a numerical code for the computation of nodes and weights low-cardinality positive quadrature formula on spherical triangles, nearly exact polynomials given degree. The algorithm is based subperiodic trigonometric Gaussian planar elliptical sectors Caratheodory–Tchakaloff compression via NNLS.

Under some non-degeneracy condition we show that sequences of entropy solutions of a nonlinear ultra-parabolic equation are strongly pre-compact in the general case of a Caratheodory flux vector. The proofs are based on localization principles for H-measures corresponding to sequences of measure-valued functions.

Let G and f : [0,1]×R4 → R be two functions satisfying Caratheodory conditions. This paper is concerned with the problems of existence and uniqueness of solutions for the nonlinear fourth-order ordinary differential equation y′′′′ +λy′′ +ky+G(x,y,y′,y′′,y′′′)= f (x,y,y′,y′′,y′′′) with one of a particular set of boundary conditions.

In this paper we introduce certain analytic functions of boundary rotation bounded by k1r which are of Caratheodory origin. With them we study two classes of analytic and univalent functions in the unit disk E = {z E C Izi < I} , which are also of bounded boundary rotation.