AI 中文总结
本文通过矩阵化简为单参数持续模的等距定理提供证明，发展了持续模的矩阵计算方法，实现了函子性持续同调条形码及特定条件下的同调条形码与态射的计算。
AI 中文摘要
本文的主要目的是为单参数持续模的等距定理提供一个简单证明，其中对交错（interleaving）的一个态射所对应的矩阵进行化简，其主元（pivots）确定了条形码（barcodes）之间的匹配关系。该方法适用于以实数为索引的有限型持续模，之后可利用q-驯服（q-tame）模的近似从上述结果推导出更一般的q-驯服模的相关结论。这种矩阵化简的使用方式与持续同调算法中矩阵化简的相似性，推动了持续模矩阵计算的进一步发展，其形式化为一个条形码范畴，其中的态射是矩阵的等价类。本文还提供了一种利用自然适配于持续同调计算的矩阵运算来计算持续同调上诱导映射的方法，使得函子性的持续同调条形码可计算；同时给出了当链未必具有无限死亡时间时，计算持续同调条形码及态射的方法。
英文摘要
The main purpose of this paper is to provide a simple proof of the isometry theorem for one-parameter persistence modules, in which a matrix representing one morphism of an interleaving is reduced and the pivots determine a matching between barcodes. This approach applies to persistence modules of finite type indexed by the reals, and the more general statement for q-tame modules can then be deduced from it using approximations of q-tame modules. The similarity between this use of matrix reduction and that in the persistent homology algorithm motivates some further development of matrix computations for persistence modules, formalized by a category of barcodes in which the morphisms are equivalence classes of matrices. A method for computing induced maps on persistent homology is provided using matrix operations that fit naturally into persistent homology computations, making functorial persistent homology barcodes computable. A method is also given for computing persistent homology barcodes and morphisms when chains do not necessarily have infinite death times.
