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Persistent and Zigzag Homology: A Matrix Factorization Viewpoint [article]

Gunnar Carlsson, Anjan Dwaraknath, Bradley J. Nelson
2021 arXiv   pre-print
Two fundamental computations in this field are persistent homology and zigzag homology.  ...  In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using  ...  Acknowledgements We thank Jonathan Taylor for extensive comments and suggestions. Conflict of interest The authors declare that they have no conflict of interest. References  ... 
arXiv:1911.10693v2 fatcat:b4lv6kt6xnfkzoiohwpxjbq2uy

The persistent homology of a sampled map: From a viewpoint of quiver representations [article]

Hiroshi Takeuchi
2019 arXiv   pre-print
Our definition of homology induced maps is given by most persistent direct summands of representations, and the direct summands uniquely determine a persistent homology.  ...  This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps.  ...  Escolar for important questions about the well-definedness of the persistence analysis. I would like to thank Zin Arai for his constructive suggestions and comments on the paper.  ... 
arXiv:1810.11774v2 fatcat:idk3niz2wrdovctam3zmxy6gqu

The persistent homology of a sampled map: from a viewpoint of quiver representations

Hiroshi Takeuchi
2021 Journal of Applied and Computational Topology  
Our definition of homology induced maps is given by most persistent direct summands of representations. The direct summands uniquely determine a persistent homology.  ...  AbstractThis paper is intended to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps.  ...  Decomposition of the zigzag persistence module as a representation yields a persistence diagram again, where each interval captures the persistence of a homology generator in the deformations of spaces  ... 
doi:10.1007/s41468-021-00065-3 fatcat:szyc3q2ofvawpgota7djmoifru

Spatio-temporal Persistent Homology for Dynamic Metric Spaces [article]

Woojin Kim, Facundo Memoli
2019 arXiv   pre-print
Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS.  ...  Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere.  ...  In [46, 47] , the thread of ideas in [11] is blended with ideas in zigzag persistence theory [14] .  ... 
arXiv:1812.00949v4 fatcat:sfscv666u5hsbdib2dlalv2ywm

Persistent homology and Floer–Novikov theory

Michael Usher, Jun Zhang
2016 Geometry and Topology  
We moreover prove a continuity result which is a natural analogue both of the classical bottleneck stability theorem in persistent homology and of standard continuity results for spectral invariants, and  ...  In the case of classical Morse theory these coincide with the barcodes familiar from persistent homology.  ...  These works are based around the notion of the (zigzag) persistent homology of level sets of the function; this is a rather different viewpoint from ours, as in order to obtain insight into Floer theory  ... 
doi:10.2140/gt.2016.20.3333 fatcat:repiha7a7fhinkmal3coevd5xe

Parallel decomposition of persistence modules through interval bases [article]

Alessandro De Gregorio, Marco Guerra, Sara Scaramuccia, Francesco Vaccarino
2021 arXiv   pre-print
We subsequently provide a parallel algorithm to build a persistent homology module over ℝ by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the  ...  This construction works for general persistence modules on a field 𝔽, not necessarily deriving from persistent homology.  ...  Acknowledgements The authors acknowledge the support from the Italian MIUR Award "Dipartimento di Eccellenza 2018-2022" -CUP: E11G18000350001 and the SmartData@PoliTO center for Big Data and Machine Learning  ... 
arXiv:2106.11884v1 fatcat:343bjbaanna4holjhsbjhj55qu

Generalized Persistence Diagrams for Persistence Modules over Posets [article]

Woojin Kim, Facundo Memoli
2020 arXiv   pre-print
By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces.  ...  This leads to a promotion of Patel's semicontinuity theorem about type 𝒜 persistence diagram to Lipschitz continuity theorem for the category of sets.  ...  Acknowledgement The idea of studying the map from the limit to the colimit of a given diagram stems from work by Amit Patel and Robert MacPherson circa 2012.  ... 
arXiv:1810.11517v5 fatcat:g6hm5dglwnfjjidcgtnsd7qlue

An Introduction to Multiparameter Persistence [article]

Magnus Bakke Botnan, Michael Lesnick
2022 arXiv   pre-print
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology.  ...  However, a single filtered space often does not adequately capture the structure of interest in the data, and one is led to consider multiparameter persistence, which associates to the data a space equipped  ...  We especially thank our past and current collaborators in this area, who have had a major influence on this article.  ... 
arXiv:2203.14289v1 fatcat:ylrfxfdljrcidondwsiiqytea4

The Jones Polynomial – Diagrams and Categories [article]

Louis H Kauffman
2022 arXiv   pre-print
In each case the Jones invariant appears as a key example for patterns and connections of these mathematical and physical contexts.  ...  This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work.  ...  a slightly different viewpoint [44] .  ... 
arXiv:2204.12104v1 fatcat:7m7tcxeshbczdov3uycoqztfum

Persistence Modules on Commutative Ladders of Finite Type [article]

Emerson G. Escolar, Yasuaki Hiraoka
2015 arXiv   pre-print
A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander-Reiten theory is applied to develop the theoretical and algorithmic foundations.  ...  Furthermore, a generalization of persistence diagrams is introduced by using Auslander-Reiten quivers.  ...  Acknowledgments The authors would like to thank Hideto Asashiba, Hiroyuki Ochiai, and Dai Tamaki for valuable discussions and comments.  ... 
arXiv:1404.7588v2 fatcat:k2lrlz4ekzafrh3roegfore5ka

Comparing Graphs via Persistence Distortion [article]

Tamal K. Dey, Dayu Shi, Yusu Wang
2017 arXiv   pre-print
This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem.  ...  In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea.  ...  Below we only provide an intuitive and informal description of the persistent homology induced by a function under our simple setting.  ... 
arXiv:1503.07414v4 fatcat:5fzvzkot3ndnjkklgmzeiw2fpi

Hair curvature: a natural dialectic and review

Joseph N. Nissimov, Asit Baran Das Chaudhuri
2014 Biological Reviews  
Zigzag hair formation has been specifically related to the expression of Dickkopf homolog 4 (Dkk4), an EDAR target gene and a Wnt inhibitor (Sharov et al., 2006; Cui et al., 2010; see also Section V.25  ...  Gene products of KRT-IA (K31, K33a, b and K34) and KRT-IIA (K81, K83 and K86) are highly homologous and are expressed in the hair cortex.  ... 
doi:10.1111/brv.12081 pmid:24617997 fatcat:qiyajpaynjhv3mhrsbhlaz5jle

Carbon Nanostructures by Macromolecular Design – From Branched Polyphenylenes to Nanographenes and Graphene Nanoribbons

Zijie Qiu, Akimitsu Narita, K. Muellen
2020 Faraday discussions  
Nanographenes (NGs) and graphene nanoribbons (GNRs) are unique connectors between the domains of 1D-conjugated polymers and 2D-graphenes.  ...  Acknowledgements The authors extend cordial thanks to all their collaborators as well as current and previous co-workers in their group who contributed to some of the results described in this article.  ...  uorescence quenching, a typical problem for large aromatic compounds, was observed, solid-state emission and emission were achieved by using a polymer matrix.  ... 
doi:10.1039/d0fd00023j pmid:33290471 fatcat:x376hqed2zhqlkjgu3gxspqrbi

Networks beyond pairwise interactions: Structure and dynamics

Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Latora, Maxime Lucas, Alice Patania, Jean-Gabriel Young, Giovanni Petri
2020 Physics reports  
We conclude with a summary of empirical applications, and an outlook on current modeling and conceptual frontiers.  ...  However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can enhance our modeling capacities and help us understand and predict their dynamical  ...  Other measures of shape in simplicial complexes Homology in all its variants (persistence, zigzag, multiparameter) is a powerful tool to classify structure according to key mesoscale features.  ... 
doi:10.1016/j.physrep.2020.05.004 fatcat:pkugz4i5obainmcrwod6i4od7q

Analytic combinatorics in several variables

2014 ChoiceReviews  
The following example may seem a natural candidate for the transfer matrix method, but it is simpler to analyze it as from the viewpoint of compositions.  ...  9.5.13 from the viewpoint of coalsecing saddles.  ...  Checking where H and H z have a common factor, we find an algebraic condition on u and v that appears to have no positive solutions.  ... 
doi:10.5860/choice.51-3299 fatcat:phaza5ah5vbmxemh2mmtchrr4u
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