标题： 整数上的广义样条函数的延拓 显示英文标题

标题： Extending Generalized Splines Over The Integers

摘要： 设 $R$ 为带有单位元的交换环，$G$ 为一个图。 \emph{扩展广义样条}在$G$上是一种顶点标记$f \in \prod_{v} M_v$，使得在每条边$e=uv$上存在一个$R$-模$M_{uv}$，以及对于边$e$的每个与之关联的顶点$u, v$都有同态$ \varphi_u : M_u \to M_{uv}$和$ \varphi_v : M_v \to M_{uv}$，满足 $\varphi_u(f_u)=\varphi_v(f_v).$扩展广义样条是广义样条的进一步推广。 它们还可以被视为模上的广义样条。本文的主要目标是研究扩展的广义样条的 $R$-模结构。我们集中于以下两个问题：一般样条的结果有多少可以推广到模上的广义样条，以及是否存在特殊基类（称为流上基）的算法或明确公式，用于模上的广义样条。我们证明了某些关于广义样条的结果可以推广到每个顶点 $v$ 被分配一个模 $M_v=m_v\mathbb Z$ 的情形。我们提供了一种算法来构造路径上这些模上的广义样条的特殊基。此外，我们引入了一种新技术，利用路径上的算法思想，在任意图上构建流上基。 显示英文摘要

摘要： Let $R$ be a commutative ring with identity and $G$ a graph. \emph{An extending generalized spline} on $G$ is a vertex labeling $f \in \prod_{v} M_v$ such that at each edge $e=uv$ there is an $R$-module $M_{uv}$ together with homomorphisms $ \varphi_u : M_u \to M_{uv}$ and $ \varphi_v : M_v \to M_{uv}$ for each vertex $u, v$ incident to the edge $e$ so that $\varphi_u(f_u)=\varphi_v(f_v).$ Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules. The main goal of this paper is to study the $R$-module structure of extending generalized splines. We concentrate on two following questions: which of the results for general splines extend to generalized splines over modules and if there is an algorithm or an explicit formula for special basis classes, called a flow up basis, for generalized splines over modules. We show that certain results concerning generalized splines can be extended to a setting where each vertex $v$ is assigned a module $M_v=m_v\mathbb Z$. We provide an algorithm to construct a special basis for generalized splines over these modules on paths. Additionally, we introduce a new technique to construct a flow-up basis on arbitrary graphs using the idea of an algorithm on paths.