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Persistent Mayer Dirac.

Faisal Suwayyid1,2, Guo-Wei Wei2,3,4

  • 1Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.

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Summary
This summary is machine-generated.

This study introduces Mayer Dirac operators for N-chain complexes, generalizing classical methods for topological data analysis. These operators enhance molecular representations and data science applications.

Keywords:
Mayer DiracMayer LaplacianMayer homologyN-chain complexbiology modelingpersistent homologytopological signals

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Area of Science:

  • Mathematics
  • Topology
  • Data Science

Background:

  • Topological data analysis (TDA) utilizes Dirac operators for signal and molecular representation.
  • Current TDA methods are limited to classical chain complexes.

Purpose of the Study:

  • To establish Mayer Dirac operators based on N-chain complexes.
  • To generalize classical Dirac operators and Laplacians for broader applications.

Main Methods:

  • Development of Mayer Dirac operators for N-chain complexes.
  • Formulation of Laplacians for N-chain complexes induced by vertex sequences.
  • Introduction of weighted Mayer Laplacian and Dirac operators.
  • Generalization of Laplacian factorization.

Main Results:

  • Established Mayer Dirac operators as a generalization of classical operators.
  • Introduced weighted operators for enhanced applicability in capturing physical attributes.
  • Demonstrated factorization of Laplacian operators.
  • Successfully applied persistent Mayer Dirac operators to biological and chemical data.

Conclusions:

  • Mayer Dirac operators offer a generalized framework for TDA.
  • Weighted operators and extensions improve practical applicability.
  • The methods show significant potential in molecular structure analysis and data science.