论文标题
Volterra积分方程及其操作员线性的代数研究
An algebraic study of Volterra integral equations and their operator linearity
论文作者
论文摘要
对特殊整体操作员的代数研究导致了Rota-Baxter操作员和Shuffle产品的概念，这些概念发现了广泛的应用。本文对一般积分运算符和方程式进行了代数研究，并表明Volterra积分运算符和相应的方程式存在着丰富的代数结构。第一伏特拉积分运算符可产生匹配的扭曲的旋转式代数代数满足扭曲的划分划分的曲线操作员身份。为了提供一个通用的空间来表达通用积分方程，然后根据包围的单词和扎根的树木构建自由操作的代数，并在顶点和边缘上装饰。匹配的扭曲旋转式代数代数中的自由对象的进一步显式结构是通过扭曲和装饰的混合产品的概括获得的，为可分离的Volterra方程提供了通用的空间。作为这些代数构建体的应用，可以证明任何具有可分离Volterra内核的积分方程都是算子线性的，因为可以将方程式简化为迭代积分的线性组合。
The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications. This paper carries out an algebraic study of general integral operators and equations, and shows that there are rich algebraic structures underlying Volterra integral operators and the corresponding equations. First Volterra integral operators are shown to produce a matching twisted Rota-Baxter algebra satisfying twisted integration-by-parts operator identities. In order to provide a universal space to express general integral equations, free operated algebras are then constructed in terms of bracketed words and rooted trees with decorations on the vertices and edges. Further explicit constructions of the free objects in the category of matching twisted Rota-Baxter algebras are obtained by a twisted and decorated generalization of the shuffle product, providing a universal space for separable Volterra equations. As an application of these algebraic constructions, it is shown that any integral equation with separable Volterra kernels is operator linear in the sense that the equation can be simplified to a linear combination of iterated integrals.
