多元Riemann可积函数的特征与完备化

给出了多元Riemann可积函数的基本特征,证明了多元Riemann可积函数空间的完备化是Lebesgue积分空间.     

G可积函数的Lebesgue可测性

Botsko在连续和可导的知识基础上推广了Riemann积分,得到了一种新的积分,称为G积分．G积分既不同于Riemann积分也不同于Lebesgue积分．本文通过对G积分的研究,得到了G可积函数一定Lebesgue可测,从而有界G可积函数一定Lebesgue可积;同时我们还证明了这两个积分值相等．     

关于从闭区间到完备随机赋范模的抽象值函数的Riemann 可积性的进一步研究

郭铁信和张霞最近引入和研究了从一个闭区间到一个完备随机赋范模的抽象值函数的Riemann积分, 证明了值域几乎处处有界的连续函数是Riemann 可积的. 本文首先给出该结果的一个更简短的证明, 使得我们对于值域的几乎处处有界性有一个更深的认识, 受此启发, 我们进一步构造两个例子, 其一说明值域并非几乎处处有界的连续函数也可以是Riemann 可积的, 另一例子说明连续函数可以非Riemann 可积. 最后, 我们证明从一闭区间到一个满支撑的完备随机赋范模的所有连续函数都Riemann 可积的充要条件是基底概率空间本质上由至多可数原子生成.     

四元数分析中k-左正则函数的性质及其Riemann边值问题

讨论了四元数分析中k-左正则函数的若干函数论性质,如Cauchy-Pompeiu公式,Cauchy公式,k-左正则函数的表示,Plemelj公式等.同时考虑了k-左正则函数的Riemann边值问题,通过k-左正则函数的Plemelj公式,将问题转化为奇异积分方程组,再利用积分方程理论和压缩映像原理证明了该问题解的存在唯一性.     

Riemann Zeta函数的延拓表达式与部分零点近似值

利用围道积分法和Riemann Zeta函数的函数方程给出了Riemann Zeta函数的另一种积分表达式,该表达式可以将Riemann Zeta函数延拓到指定的右半平面.利用该表达式求出了ζ(2n)、ζ(1-2n)和ζ’(0),并且计算了Riemann Zeta函数非平凡零点的部分数值解.该积分表达式的引出丰富了与Riemann Zeta函数延拓表达式相关问题的研究.     

二维双曲守恒律的大时间步长Godunov方法(英文)

考虑了关于二维守恒律的大时间步长Godunov方法.该方法是关于一维问题的自然推广.证明了文中给出的数值流函数下,该方法是守恒的.进一步还给出了近似Riemann解应满足的条件,并且证明了利用满足这些条件的近似Riemann解的大时间步长Godunov方法守恒.最后,补充证明了满足这些条件的近似Riemann解是满足熵条件的.     

平面上同分母有理点的稠密性及其应用

证明了实平面上同分母有理点集的稠密性.作为此结果的应用,构造了实平面上的一个简单函数,此函数说明实平面上具有稠密不连续点集的Riemann可积函数是存在的.     

满足同一个Riemann型函数方程的L-函数的唯一性

利用L-函数的性质证明了满足同一个Riemann型函数方程的扩充的Selberg类中的次数大于零的L-函数分担一个有限值的定理,并应用该定理证明了满足同一个Riemann型函数方程的涉及公共值的唯一性定理,所得结果改进了J.Steuding和李宝勤的有关结果,也是S.M.Gonek,J.Haan和H.Ki主要结果的补充.     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

黎曼积分的完备化

综述了黎曼可积函数的基本特征,并指出黎曼可积函数列的极限运算在积分意义下是不封闭的.在构造了完备化空间之后,证明了该空间就是勒贝格可积函数空间,从而说明了黎曼积分的完备化形式是勒贝格积分.     

On the first-return integrals

Some pathological properties of the first-return integrals are explored. In particular it is proved that there exist Riemann improper integrable functions which are first-return recoverable almost everywhere, but not first-return integrable, with respect to each trajectory. It is also proved that the usual convergence theorems fail to be true for the first-return integrals.     

Invariant Metrizability and Projective Metrizability on Lie Groups and Homogeneous Spaces

In this paper, we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

     

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.     

Fractional Integrals on Variable Hardy–Morrey Spaces

Vector-valued fractional maximal inequalities on variable Morrey spaces are proved. Applying atomic decomposition of variable Hardy–Morrey spaces, we obtain the boundedness of fractional integrals on variable Hardy–Morrey spaces, which extends the Taibleson–Weiss’s results for the boundedness of fractional integrals on Hardy spaces. The corresponding boundedness for the fractional type integrals is also considered.     

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L¹ or the real Hardy spaces defined on ℝⁿ, where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H¹(ℝ) into L¹(ℝ) and from L¹(ℝ) into weak –L¹(ℝ). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H^(1,0)(ℝIsup2), H^(0,1)(ℝ²⁾ into weak –L¹(ℝ²). Hence pointwise Riemann summability of Fourier integrals of functions in H^(1,0) ∪ H^(0,1)(ℝ²) follows almost everywhere.The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.     

Typical dimension of the graph of certain functions

Most functions from the unit interval to itself have a graph with Hausdorff and lower entropy dimension 1 and upper entropy dimension 2. The same holds for several other Baire spaces of functions. In this paper it will be proved that this is the case also in the spaces of all mappings that are Lebesgue measurable, Borel measurable, integrable in the Riemann sense, continuous, uniform distribution preserving (and continuous).     

csf可数空间的注记

讨论csf可数空间的性质,把csf可数空间刻画为度量空间的映像,同时探讨了伪紧的csf可数空间的第一可数性质,推广了Arhangel’skiˇ?关于度量空间伪开s映像的结果,证明了正则伪紧的仿拓扑群是可度量化的当且仅当它是csf可数的Fr′echet空间.     

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.     

Stochastic integrals in Riemann manifolds

Stochastic integrals are constructed with values in a compact Riemann manifold from a continuous martingale integrator that is given in the tangent space of the initial point of the stochastic integral and from a stochastic tensor field of linear endomorphisms of the tangent bundle. The integrals that are formed are continuous processes that suitably preserve the martingale property. These stochastic integrals should be useful for the applications of a stochastic calculus in Riemann manifolds.     

Morrey-Type Spaces on Gauss Measure Spaces and Boundedness of Singular Integrals

In this paper, the authors introduce Morrey-type spaces on the locally doubling metric measure spaces, which means that the underlying measure enjoys the doubling and the reverse doubling properties only on a class of admissible balls, and then obtain the boundedness of the local Hardy–Littlewood maximal operator and the local fractional integral operator on such Morrey-type spaces. These Morrey-type spaces on the Gauss measure space are further proved to be naturally adapted to singular integrals associated with the Ornstein–Uhlenbeck operator. To be precise, by means of the locally doubling property and the geometric properties of the Gauss measure, the authors establish the equivalence between Morrey-type spaces and Campanato-type spaces on the Gauss measure space, and the boundedness for a class of singular integrals associated with the Ornstein–Uhlenbeck operator (including Riesz transforms of any order) on Morrey-type spaces over the Gauss measure space.