Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Functions not first-return integrable

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0,1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C^∞ on (0,1] yet fails to be first-return integrable.     

Stability Analysis of nth Order Nonlinear Impulsive Differential Equations in Quasi–Banach Space

Abstract

This article is about Ulam’s type stability of n^th order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam’s type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder’s inequality and the beta integral inequality.     

黎曼相关积分研究

通过证明和反例讨论黎曼积分、直接黎曼积分、黎曼-斯蒂尔切斯积分三者间的联系与区别.结果显示：若函数直接黎曼可积,则它黎曼可积,并且两者积分值相同,但反之不成立;若函数黎曼可积,则任意连续函数关于该函数不一定黎曼-斯蒂尔切斯可积.从讨论结果中还获得直接黎曼可积和黎曼可积各自的一个充分条件.     

On Double Stratonovich Fractional Integrals and Some Strong and Weak Approximations

Abstract

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived.

Riemann–Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.     

Almost every sequence integrates

The purpose of this paper is to discuss a first-return integration process which yields the Lebesgue integral of a bounded measurable function f: I → R defined on a compact interval I. The process itself, which has a Riemann flavor, uses the given function f and a sequence of partitions whose norms tend to 0. The “first-return” of a given sequence is used to tag the intervals from the partitions. The main result of the paper is that under rather general circumstances this first return integration process yields the Lebesgue integral of the given function f for almost every sequence . This research was initiated while the authors were in residence at the Mathematical Institute of St. Andrews University.     

On Ito-Kurzweil-Henstock Integral and Integration-by-Part Formula

In this paper we derive the Integration-by-Parts Formula using the generalized Riemann approach to stochastic integrals, which is called the Ito-Kurzweil-Henstock integral.     

Consistent first-return Riemann sums for Lebesgue integrals

U. B. Darji and M. J. Evans [1] showed previously that it is possible to obtain the integral of a Lebesgue integrable function on the interval [0,1] via a Riemann type process, where one chooses the selected point in each partition interval using a first-return algorithm based on a sequence {x _n} which is dense in [0,1]. Here we show that if the same is true for every rearrangement of {x _n}, then the function must be equal almost everywhere to a Riemann integrable function. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimization of Functional Integral Equations: A Spectral Approach

Abstract

The spectral method of Elnagar and Kazemi (J. Comp. Appl. Math. 76(1–2):147–158, 1996), which yields spectral convergence rate for the approximate solutions of Volterra-Hammerstein integral equations, is generalized in order to solve the larger class of functional integral equation control systems with spectral accuracy. The proposed method is based on the idea of relating spectrally constructed grid points to the structure of projection operators. These operators will be used to approximate the control vector and the associated state vector. Numerical examples are included to demonstrate the accuracy of the proposed method.     

无穷限积分的控制收敛理论

本文根据有限区间上Ｒｉｅｍａｎｎ积分的Ａｒｚｅｌａ控制收敛定理［１］，给出无穷限积分的控制收敛定理，并做了相应的推广。     

Henstock-Kurzweil and McShane product integration; Descriptive definitions

The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function A is absolutely continuous. As a consequence we obtain that the McShane product integral of A over [a, b] exists and is invertible if and only if A is Bochner integrable on [a, b]. Supported by grant No. 201/04/0690 of the Grant Agency of the Czech Republic.     

Meromorphic Functions on a Riemann Surface to a Locally Semi-convex Space

In this article, vector-valued holomorphic and meromorphic functions on a Riemann surface to a complete Hausdorff locally semi-convex space are discussed. By introducing the concepts of vector-valued holomorphic and meromorphic differential forms, Cauchy's theorem and the Residue theorem of a vector-valued differential form on a Riemann surface are proved. Using the theory on the operator and the theory of a cohomology of a sheaf, we give a proof of the Mittag-Leffler theorem for vector-valued meromorphic functions on a non-compact Riemann surface to a complete Hausdorff locally semi-convex space.     

Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s,?m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.     

On the Hartogs-Phenomenon and Extension of Analytic Hypersurfaces in Non-separated Riemann Domains

In the present paper, we answer two questions raised by Jarnicki and Pflug: First, we show by a counterexample that the Hartogs-Bochner theorem is no longer true for non-separated Riemann domains. Secondly, we generalize a structure theorem of Dloussky, which examines the extension of singularity sets contained in analytic hypersurfaces, to non-separated Riemann domains. Moreover, our method yields a new proof of Dloussky's original result.     

Energy and Foliations on Riemann Surfaces

We define the energy of foliations on Riemann surfaces. We prove that meromorphic vector fields are critical points and we compute their energies using the Green’s function. We then generalize the results to principal circle bundles over Riemann surfaces.Mathematics Subject Classification (2000): 53C12, 53C15.     

THE LIMIT SET OF RIEMANN SUMS OF A VECTOR STIELTJES INTEGRAL

Abstract

Conditions for convexity of the limit set of Riemann sums of a vector integral are investigated.     

Centralizers of Rational Functions

We show that the elements of an open and dense subset of rational functions of the Riemann sphere have trivial centralizers; i.e, the rational functions commute only with their own powers.     

Minimal surfaces in R3 with one end¶and bounded curvature

We study complete minimal surfaces M immersed in R ³, with finite topology and one end. We give conditions which oblige M to be conformally a compact Riemann surface punctured in one point, and we show that M can be parametrized by meromorphic data on this compact Riemann surface. The goal is to prove that when M is also embedded, then the end of M is asymptotic to an end of a helicoid (or M is a plane). Received: 13 January 1997 / Revised version: 15 September 1997     

A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type

Abstract

We show the existence of weak solutions in an elliptic region in the self-similar plane to the two-dimensional Riemann problem for the pressure-gradient system of the compressible Euler system. The two-dimensional Riemann problem we study is the interaction of two forward rarefaction waves, which are adjacent to a common vacuum that occupies a sectorial domain of 90 degrees. We assume the origin is on the boundary of the domain. In addition, the domain is open, bounded, and simply connected with a piecewise C ^2,α boundary. We resolve the difficulty that arises from the fact that the origin is on the boundary of the domain.     

Improper Riemann Integral and Henstock Integral in ℝn

The Henstock integral in ℝⁿ and its relation to the n-dimensional improper Riemann integral are studied. A Hake-type theorem for the Henstock integral in ℝⁿ is proved.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 251–258.Original Russian Text Copyright © 2005 by P. Muldowney, V. A. Skvortsov.