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Ullah, Md. Safik
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Education Mathematics Social Sciences

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A review of some properties of persistent homology Ullah, Md. Safik; Mou, Mst. Sima Akhter; Shiraj, Md. Morshed Bin; Rahman, Md. Mizanur; Murshed, Md. Masum; Akhter, Nasima
Desimal: Jurnal Matematika Vol. 6 No. 2 (2023): Desimal: Jurnal Matematika
Publisher : Universitas Islam Negeri Raden Intan Lampung

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24042/djm.v6i2.17812

Abstract

Every day, enormous complex geometric data are accumulating rapidly, and qualitative analysis is needed, which cannot be done properly without studying the shapes of those data. Persistent homology describes the homology of data sets of arbitrary size, producing state-of-the art results in data analysis across a significant number of fields and sparking a rigorous study of persistence in homology theory. In this study, persistent homology has been demonstrated as a homology theory by satisfying the Eilenberg-Steenrod axioms. A brief background on persistent homology groups has been written to understand their construction. Then other definitions of persistent homology based on functors and graded modules have also been reviewed. The Mayer-Vietoris-Vietorisfor persistent homology has been derived as a property of persistent homology. Subsequently, a long, exact sequence for persistent homology has been constructed. Furthermore, the stability of persistent homology has been examined carefully. Finally, the Diamond principle of persistent homology has been explained briefly. This study can be used to investigate new properties of persistent homology, among other benefits.
Co-Authors Akhter, Nasima Md. Mizanur Rahman Mou, Mst. Sima Akhter Murshed, Md. Masum Shiraj, Md. Morshed Bin