Suwayyid Faisal, Wei Guo-Wei
Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.
Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States of America.
J Phys Complex. 2024 Dec 1;5(4):045005. doi: 10.1088/2632-072X/ad83a5. Epub 2024 Oct 17.
Topological data analysis (TDA) has made significant progress in developing a new class of fundamental operators known as the Dirac operator, particularly in topological signals and molecular representations. However, the current approaches being used are based on the classical case of chain complexes. The present study establishes Mayer Dirac operators based on -chain complexes. These operators interconnect an alternating sequence of Mayer Laplacian operators, providing a generalization of the classical result . Furthermore, the research presents an explicit formulation of the Laplacian for -chain complexes induced by vertex sequences on a finite set. Weighted versions of Mayer Laplacian and Dirac operators are introduced to expand the scope and improve applicability, showcasing their effectiveness in capturing physical attributes in various practical scenarios. The study presents a generalized version for factorizing Laplacian operators as an operator's product and its 'adjoint'. Additionally, the proposed persistent Mayer Dirac operators and extensions are applied to biological and chemical domains, particularly in the analysis of molecular structures. The study also highlights the potential applications of persistent Mayer Dirac operators in data science.
拓扑数据分析（TDA）在开发一类新的基本算子（即狄拉克算子）方面取得了重大进展，特别是在拓扑信号和分子表示方面。然而，目前使用的方法是基于链复形的经典情况。本研究基于(\mathbb{Z}_2)-链复形建立了迈耶狄拉克算子。这些算子将迈耶拉普拉斯算子的交替序列相互连接起来，给出了经典结果的一个推广。此外，该研究给出了由有限集上的顶点序列诱导的(\mathbb{Z}_2)-链复形的拉普拉斯算子的显式公式。引入了迈耶拉普拉斯算子和狄拉克算子的加权版本，以扩大其范围并提高适用性，展示了它们在各种实际场景中捕捉物理属性的有效性。该研究给出了将拉普拉斯算子分解为一个算子与其“伴随”的乘积的广义版本。此外，所提出的持久迈耶狄拉克算子及其扩展被应用于生物和化学领域，特别是在分子结构分析中。该研究还强调了持久迈耶狄拉克算子在数据科学中的潜在应用。
