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ArXiv : 1502.07445.
"Homological scaffolds of brain functional networks".
See also edit References edit Epstein, Charles ; Carlsson, Gunnar ; Edelsbrunner, Herbert.
"When remote sensing meets topological data analysis"."Multidimensional Size Functions for Shape Comparison".Nakamura, Takenobu; Hiraoka, Yasuaki; Hirata, Akihiko; Escolar, Emerson.; Nishiura, Yasumasa.The VietorisRips complex is preferred over ech complex because its definition is simpler and the ech complex requires extra effort to define in a general finite metric space."Numeric Invariants from Multidimensional Persistence".A b Carlsson, Gunnar.9 Edelsbrunner and Harer's book gives general guidance on computational topology.Cagliari, Francesca; Landi, Claudia.4 In 1990, Patrizio Frosini introduced the size function, which is equivalent to the 0th persistent homology.This allows many statistical tools to be ported to TDA.

Homology, Homotopy and Applications.
It is evident to mathematicians that persistent homology is closely related to spectral sequences.
Specifically, V r ( X ) C 2 r ( X ) V 2 r ( X ) displaystyle V_r(X)subset C_sqrt 2r(X)subset V_2r(X).
Then for any two functors F, G : P D textstyle F,G:Pto D, we have d I ( H F, H G ) d I ( F, G ) textstyle d_I(HF, HG)leq d_I(F,G).Have abstracted the definition to that between functors F, G : P D displaystyle F,G:Pto D when P textstyle P is equipped with a sublinear projection or superlinear family, in which still remains a pseudometric.It covers stemming, stop words, document summarization, visualization, segmentation, categorization and clustering.In the literature, the rank invariant is often referred as the persistent Betti numbers (PBNs).Horak, Danijela; Maleti, Slobodan; Rajkovi, Milan.