In this article, we present characterizations of the concavity property minimal $$L^2$$ integrals degenerating to linearity in case products analytic subsets on open Riemann surfaces. As applications, obtain holding equality optimal jets extension problem from surfaces, which implies product versions parts Suita conjecture and extended conjecture, a Ohsawa for

This paper first reviews quantum measure and integration theory. A new representation of the quantum integral is presented. This representation is illustrated by computing some quantum (Lebesgue)2 integrals. The rest of the paper only considers finite spaces. Anhomomorphic logics are discussed and the classical domain of a coevent is studied. Pure quantum measures and coevents are considered an...

This paper provides a discrete Poincaré inequality in n space dimensions on a simplex K with explicit constants. This inequality bounds the norm of the piecewise derivative of functions with integral mean zero on K and all integrals of jumps zero along all interior sides by its Lebesgue normbyC (n)diam(K ). The explicit constantC (n) depends only on the dimension n = 2,3 in case of an adaptive ...

The theory of fuzzy measures and fuzzy integrals was introduced by Sugeno [24] as a tool for modeling nondeterministic problems. Sugeno’s integral is analogous to Lebesgue integral which has been studied by many authors, including Pap [18], Ralescu and Adams [19] and, Wang and Klir [25], among others. RománFlores et al [9, 20–23], started the studies of inequalities for Sugeno integral, and the...

Abstract. We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous ana...

The classic Lebesgue ANOVA expansion offers an elegant way to represent functions that depend on a high-dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the ANOVA representation is constructed. Unfortunately, the construction of the expansion itself is expensive due to the need to evaluate high-dimensional integrals. A way ...

1. This paper investigates certain linear differential systems whose solutions are required to be merely functions of bounded variation. Subsequently the equivalence of such problems with appropriate integral equations with Lebesgue integrals and, also, their equivalence to certain integral equations with Stieltjes integrals is established. This study is a generalization of differential systems...

Given a density d de…ned on the Borel subsets of [0;1); the limit at in…nity in density of a function f : [0;1) ! R is zero if each of the sets ft : jf(t)j "g has zero density whenever " > 0: It is proved that every Lebesgue integrable function f : [0;1) ! R veri…es this type of behavior at in…nity with respect to a scale of densities including the usual one, d(A) = limr!1 m(A\[0;r)) r : The an...

The following discussion is closely connected with Lebesgue's theorem that the derivative of an integral is equal to the integrand almost everywhere. I t is well known that in generalizing this theorem to higher dimensions, great care must be exercised in the choice of the systems of intervals or sets used for ^-dimensional differentiation. Lebesgue had already observed that arbitrary intervals...