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The authors provide important definitions and theorems on critical edges, local minima, and selective Rips complexes, and caution that certain assumptions are necessary for their results to hold. #Topology #RipsComplexes #PersistentHomology #ComputerScience #TopologicalDataAnalysis
To compute #PersistentHomology, one needs to check, for each d-simplex σi of a filtration, if it "fills" a (d-1)-dimensional hole, i.e. if its boundary ∂σi is homologous to a non-trivial (d-1)-cycle created on an unpaired (d-1)-simplex (blue).
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https://arxiv.org/abs/2206.13932
#TopologicalDataAnalysis #Visualization #DataScience
Funded by the European Research Council (ERC) (project TORI, https://erc-tori.github.io/)
Two p-cycles a and b are "homologous" (i.e. belong to the same #Homology class) if there exists a (p+1)-chain c, such that b = a + ∂c (mod-2 sum)
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https://arxiv.org/abs/2206.13932
#TopologicalDataAnalysis #TopologyToolKit #PersistentHomology #Visualization #DataScience #MachineLearning
Funded by the European Research Council (ERC) (project TORI, https://erc-tori.github.io/)
Check out our new paper with @_gosiao on the #TopologicalDataAnalysis of magnetic currents in molecules with the #TopologyToolKit 👇
Accepted to @PCCP
https://arxiv.org/abs/2212.08690
#PersistentHomology #visualization #compchem #paraview
Topological persistence is an importance measure in #PersistentHomology, with a strong practical utility for noise removal in various applications: #Imaging, #Clustering, #ClimateScience, #Geophysics, #MaterialScience and more! 👇
https://arxiv.org/pdf/2206.13932
#TopologicalDataAnalysis
