函数可积与原函数存在的关系

详细探讨函数黎曼可积性与原函数存在性之间的相互关系,通过构造具体函数说明黎曼可积与原函数存在是相互独立形成的不同概念,它们之间是互不蕴含的关系.     

Hermite函数的原函数的平方可积性

讨论了n阶Hermite函数的"变上限积分"型原函数在R=(-∞,+∞)上的平方可积性,证明了当n为偶数时这种原函数不是平方可积,而当n为奇数时这种原函数是平方可积的,并给出了n为奇数时原函数的L~2(R)范数的上界.     

分段函数、函数的可积性与原函数存在性

论述了分段函数在数学分析中的作用,并以分段函数为工具,给出了函数的原函数存在和黎曼可积之间的关系,有助于全面掌握原函数和定积分这两个重要概念.     

R可积与原函数

本文采用集合的测度概念和L积分理论,较简明地证明了函数R可积的充要条件,深入分析了月可积函数的本质特征,给出不存在原函数的R可积函数,构造反例证明了存在原函数的有界函数未必是R可积的,从两方面论证月可积函数与原函数之间的互不包含关系。     

模糊数值函数的积分原函数是否几乎处处可导

对于模糊数值函数的积分原函数的可导性问题,本文构造性地给出一反例,说明存在(K)可积的模糊数值函数其积分原函数并不是几乎处处可导的。     

Henstock引理,导函数的可积性,积分原函数的可导性

对经典实分析非绝对积分理论中的Henstock引理的本质特征进行了讨论,指出:Henstock引理的本质是刻划了导函数的可积性和积分原函数的可导性问题.     

函数的原函数存在与黎曼可积的关系

本文讨论原函数存在与黎曼可积之间的联系与区别,通过列举具体的函数来说明函数的原函数存在与黎曼可积是相互独立的概念,两者之间是互不蕴舍的关系.     

析谈单元函数积分的基本概念与基本理论

<正> 单元函数积分学包含不定积分和定积分这两部分内容,其中原函数、不定积分和定积分的概念是其基本概念,积分中值定理、上限是变量的定积分及其求导定理是其基本理论,而Newton-Lebniz公式是其基本公式,积分法是其基本的运算法.本文将侧重围绕着积分学的基本概念和基本理论,论述三个关系,即原函数与不定积分;不定积分与定积分;原函数的存在性与可积性的关系以及积分中值定理.     

利用几何意义求函数单调性

通过变量代换,使原函数具有几何意义,数形结合,观察当x变大时,动点如何变,函数值变大还是变小,便可得到原函数的单调性,请看以下三例．     

积分区域与原函数的存在性

讨论由Newton-Leibuniz公式引起的积分区域对原函数存在性的影响问题.     

Differential Galois theory and non-integrability of planar polynomial vector fields

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.     

超四类可积函数

结合实变函数理论,对数学分析中有界函数的Riemann可积性问题进行研究,在有关文献所提出的第四类可积函数的基础上,引入超四类可积函数,并以实例说明超四类可积函数是存在的．     

Dirichlet函数与Riemann函数的性质及应用

探讨Dirichlet函数和Riemann函数的连续性、可导性和可积性,并由此分别构造出在一点连续、多点连续、一点可导、多点可导的函数.     

Revisiting the problem of integrability in utility theory

We revisit the problem of integrability in the consumer theory, focusing on the main difficulties. First, we look for a neat and simple local existence result, and then for a global solution. Second, observing that a utility function (or indirect utility function) cannot be determined uniquely, we propose a means to get a kind of uniqueness result. Our approach is coordinate-free and can be used both in the classical case of a finite-dimensional commodity space and in the case an infinite-dimensional model is adopted.     

Existence results for Urysohn integral inclusions involving the Henstock integral

This paper presents two existence results for Urysohn integral inclusions in Banach spaces, the set-valued integral involved being the Henstock integral. In particular, the existence of continuous solutions for integral inclusions of Volterra and of Hammerstein type is obtained. Our results extend the existence theorems given in literature in the single- or set-valued case, under Bochner or Pettis integrability hypothesis. The compactness of solutions set is also studied.     

反函数法求定积分

由定积分的可积条件与分部积分法推出一种利用反函数求解定积分的简捷方法.     

具随机折现的博弈期权定价问题

对具随机折现的博弈期权定价问题进行了研究,在满足一个可积性条件的情况下,借用过份函数等工具给出了期权价格的表达式和买卖双方的最优停止策略.对于不满足可积性条件的情况,推广了相关文献的结果,并给出了τ＊存在的条件.最后给出了一个例子.     

Solutions to the 2D Euler equations with velocity unbounded at infinity

We prove existence of solutions to the two-dimensional Euler equations with vorticity bounded and with velocity locally bounded but growing at infinity at a rate slower than a power of the logarithmic function. We place no integrability conditions on the initial vorticity. This result improves upon a result of Serfati which gives existence of a solution to the two-dimensional Euler equations with bounded velocity and vorticity.     

MBIUS-TODA HIERARCHY AND ITS INTEGRABILITY

In this paper, we construct a new integrable equation called Mbius-Toda equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Mbius-Toda equation and a whole integrable Mbius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the Mbius-Toda hierarchy are given and this leads to the existence of the tau function.