{"title":"Simplicial branching random walks","authors":"Ron Rosenthal","doi":"10.1007/s41468-023-00148-3","DOIUrl":"https://doi.org/10.1007/s41468-023-00148-3","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"114 30","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135138044","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}
Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela
{"title":"Rigorous computation in dynamics based on topological methods for multivector fields","authors":"Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela","doi":"10.1007/s41468-023-00149-2","DOIUrl":"https://doi.org/10.1007/s41468-023-00149-2","url":null,"abstract":"Abstract Motivated by the theoretical results of Mrozek et al. (Commun Nonlinear Sci Numer Simul 108:106–226, 2022) we present an algorithmic construction of a transversal cellular decomposition for a planar ODE. We then use the associated combinatorial multivector field to algorithmically detect the existence of an isolated invariant set with the Conley index of a periodic orbit and admitting a combinatorial Poincaré section. This construction combined with the theoretical results of Mrozek et al. (2022) leads to a method for automatized computer assisted proofs of the existence of periodic solutions in ODE’s.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"28 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135774489","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}
Bruno Benedetti, Crystal Lai, Davide Lofano, Frank H. Lutz
{"title":"Random simple-homotopy theory","authors":"Bruno Benedetti, Crystal Lai, Davide Lofano, Frank H. Lutz","doi":"10.1007/s41468-023-00139-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00139-4","url":null,"abstract":"Abstract We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions . For triangulated d -manifolds with $$dle 6$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, $$(14k+1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>14</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -vertex triangulations of a new series of Bing’s houses with k rooms, $$kge 3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , which all can be deformed to a point using only six pure elementary expansions.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157610","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":"Yarn ball knots and faster computations","authors":"Dror Bar-Natan, Itai Bar-Natan, Iva Halacheva, Nancy Scherich","doi":"10.1007/s41468-023-00144-7","DOIUrl":"https://doi.org/10.1007/s41468-023-00144-7","url":null,"abstract":"Abstract We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"19 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157611","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":"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}
Samir Chowdhury, Tom Needham, Ethan Semrad, Bei Wang, Youjia Zhou
{"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}
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