优化$A_\infty$持久组的拓扑信息
Optimising the topological information of the $A_\infty$-persistence groups
摘要：
持久同调通常研究同调群$H_p(X)$（系数在域中）沿着拓扑空间的过滤的演化。$A_\infty$ -持久性通过分析子空间如$V := \text{Ker}\, {Δ_n}_{| H_p(X)} \subseteq H_p(X)$的演化扩展了这个理论，其中$\{Δ_m\}_{m\geq1}$表示$H_*(X)$上的$A_\infty$ -余代数的结构。本文通过讨论$A_\infty$ -持久群的最基本形式$V$的拓扑意义，说明了$A_\infty$-持久群在持久同调之外的作用.此外，我们还探讨了如何沿着滤子选择$A_\infty$ -余代数，以使$A_\infty$ -持久群携带更多的忠实信息。
英文摘要：
Persistent homology typically studies the evolution of homology groups $H_p(X)$ (with coefficients in a field) along a filtration of topological spaces. $A_\infty$-persistence extends this theory by analysing the evolution of subspaces such as $V := \text{Ker}\, {Δ_n}_{| H_p(X)} \subseteq H_p(X)$, where $\{Δ_m\}_{m\geq1}$ denotes a structure of $A_\infty$-coalgebra on $H_*(X)$. In this paper we illustrate how $A_\infty$-persistence can be useful beyond persistent homology by discussing the topological meaning of $V$, which is the most basic form of $A_\infty$-persistence group. In addition, we explore how to choose $A_\infty$-coalgebras along a filtration to make the $A_\infty$-persistence groups carry more faithful information.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
