Journal of applied and computational topology最新文献

{"title":"Multivariate central limit theorems for random clique complexes","authors":"Tadas Temčinas, Vidit Nanda, Gesine Reinert","doi":"10.1007/s41468-023-00146-5","DOIUrl":"https://doi.org/10.1007/s41468-023-00146-5","url":null,"abstract":"Abstract Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching, the vector of simplex counts in the link of a fixed simplex, and the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U -statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"7 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135510925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spherical coordinates from persistent cohomology","authors":"Nikolas C. Schonsheck, Stefan C. Schonsheck","doi":"10.1007/s41468-023-00141-w","DOIUrl":"https://doi.org/10.1007/s41468-023-00141-w","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finding the homology of manifolds using ellipsoids","authors":"Sara Kališnik, Davorin Lešnik","doi":"10.1007/s41468-023-00145-6","DOIUrl":"https://doi.org/10.1007/s41468-023-00145-6","url":null,"abstract":"Abstract A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded \"Equation missing\"<!-- image only, no MathML or LaTex -->-submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136354168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Persistent cup product structures and related invariants","authors":"Facundo Mémoli, Anastasios Stefanou, Ling Zhou","doi":"10.1007/s41468-023-00138-5","DOIUrl":"https://doi.org/10.1007/s41468-023-00138-5","url":null,"abstract":"Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length $$ell ge 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the other is the filtration parameter. This new persistence structure, called the persistent cup module , is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> , we obtain a 1-dimensional persistence module, called the persistent $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> -cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a persistent invariant , which extends both the rank invariant (also referred to as persistent Betti number ), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant , and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant .","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hypergraph co-optimal transport: metric and categorical properties","authors":"Samir Chowdhury, Tom Needham, Ethan Semrad, Bei Wang, Youjia Zhou","doi":"10.1007/s41468-023-00142-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00142-9","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"160 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136280508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the nodal structures of random fields: a decade of results","authors":"Igor Wigman","doi":"10.1007/s41468-023-00140-x","DOIUrl":"https://doi.org/10.1007/s41468-023-00140-x","url":null,"abstract":"Abstract We survey a decade worth of work pertaining to the nodal structures of random fields, with emphasis on the transformative techniques that shaped the field.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136279525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large simplicial complexes: universality, randomness, and ampleness","authors":"Michael Farber","doi":"10.1007/s41468-023-00134-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00134-9","url":null,"abstract":"Abstract The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality (Even-Zohar et al. in Eur J Math 8(1):1–32, 2022; Farber and Mead in Topol Appl 272(22):107065, 2020; Farber et al. in J Appl Comput Topol 5(2):339–356, 2021). In the first part of the paper we concentrate on r -ample simplicial complexes which are high dimensional analogues of the r -e.c. graphs introduced originally by Erdős and Rényi (Acta Math Acad Sci Hungar 14:295–315, 1963), see also Bonato (Contrib Discrete Math 4(1):40–53, 2009). The class of r -ample complexes is useful for applications since these complexes allow extensions of subcomplexes of certain type in all possible ways; besides, r -ample complexes exhibit remarkable robustness properties. We discuss results about the existence of r -ample complexes and describe their probabilistic and deterministic constructions. The properties of random simplicial complexes in medial regime (Farber and Mead 2020) are important for this discussion since these complexes are ample, in certain range. We prove that the topological complexity of a random simplicial complex in the medial regime satisfies $$textsf{TC}(X)le 4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>TC</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , with probability tending to 1 as $$nrightarrow infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . There exists a unique (up to isomorphism) $$infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>∞</mml:mi> </mml:math> -ample complex on countable set of vertexes (the Rado complex), and the second part of the paper surveys the results about universality, homogeneity, indestructibility and other important properties of this complex. The Appendix written by J.A. Barmak discusses connectivity of conic and ample complexes.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"133 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135306542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Representing fields without correspondences: the lifted Euler characteristic transform","authors":"Henry Kirveslahti, S. Mukherjee","doi":"10.1007/s41468-023-00133-w","DOIUrl":"https://doi.org/10.1007/s41468-023-00133-w","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53194506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Vietoris–Rips complexes of metric spaces near a metric graph","authors":"S. Majhi","doi":"10.1007/s41468-023-00122-z","DOIUrl":"https://doi.org/10.1007/s41468-023-00122-z","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"7 1","pages":"741 - 770"},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46073955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the homology language of HDA models of transition systems","authors":"Thomas Kahl","doi":"10.1007/s41468-023-00120-1","DOIUrl":"https://doi.org/10.1007/s41468-023-00120-1","url":null,"abstract":"Abstract Given a transition system with an independence relation on the alphabet of labels, one can associate with it a usually very large symmetric higher-dimensional automaton. The purpose of this paper is to show that by choosing an acyclic relation whose symmetric closure is the given independence relation, it is possible to construct a much smaller nonsymmetric HDA with the same homology language.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136375477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}