Pub Date : 2023-09-30DOI: 10.1007/s41468-023-00142-9
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":null,"pages":null},"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}
Pub Date : 2023-09-30DOI: 10.1007/s41468-023-00140-x
Igor Wigman
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.
摘要:我们调查了十年来关于随机场节点结构的工作，重点是塑造该领域的变革技术。
{"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":null,"pages":null},"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}
Pub Date : 2023-09-16DOI: 10.1007/s41468-023-00134-9
Michael Farber
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 $$textsf{TC}(X)le 4$$ TC(X)≤4 , with probability tending to 1 as $$nrightarrow infty $$ n→∞ . There exists a unique (up to isomorphism) $$infty $$ ∞ -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.
摘要本文综述了近年来在理解大型简单复合体的几何、拓扑和组合性质方面的研究进展，主要集中在丰度、连通性和普遍性方面(Even-Zohar et al. in Eur J Math 8(1):1 - 32,2022;中国生物医学工程学报，32 (2):444 - 444;Farber et al. [J] .计算机学报(英文版)5(2):339-356,2021)。在本文的第一部分，我们将重点放在r -样例简单复形上，它们是最初由Erdős和r nyi(匈牙利数学学术学报，1963年14:295-315)引入的r -e.c.图的高维类似物，也参见Bonato(贡献离散数学4(1):40 - 53,2009)。r -样本配合物类在应用中是有用的，因为这些配合物允许以所有可能的方式扩展特定类型的子配合物;此外，r -ample配合物具有显著的鲁棒性。讨论了r样本复合体存在性的结果，并描述了它们的概率和确定性结构。随机简单复合物在药物治疗中的性质(Farber and Mead 2020)对于本讨论很重要，因为这些复合物在一定范围内是丰富的。我们证明了一个随机简单复体在中间区域的拓扑复杂度满足$$textsf{TC}(X)le 4$$ TC (X)≤4，其概率趋于1为$$nrightarrow infty $$ n→∞。在可数顶点集上存在一个唯一(直至同构)$$infty $$∞-样复数(Rado复形)，本文第二部分讨论了该复形的普适性、同质性、不可破坏性等重要性质。由J.A. Barmak撰写的附录讨论了圆锥和充裕复合体的连通性。
{"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":null,"pages":null},"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}
Pub Date : 2023-08-04DOI: 10.1007/s41468-023-00133-w
Henry Kirveslahti, S. Mukherjee
{"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":null,"pages":null},"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}
Pub Date : 2023-06-17DOI: 10.1007/s41468-023-00126-9
R. Biswas, Sebastiano Cultrera di Montesano, H. Edelsbrunner, M. Saghafian
{"title":"Geometric characterization of the persistence of 1D maps","authors":"R. Biswas, Sebastiano Cultrera di Montesano, H. Edelsbrunner, M. Saghafian","doi":"10.1007/s41468-023-00126-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00126-9","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44570458","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}
Pub Date : 2023-05-15DOI: 10.1007/s41468-023-00122-z
S. Majhi
{"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":null,"pages":null},"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}
Pub Date : 2023-05-04DOI: 10.1007/s41468-023-00120-1
Thomas Kahl
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.
{"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":null,"pages":null},"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}
Pub Date : 2023-03-17DOI: 10.1007/s41468-023-00116-x
J. Boissonnat, M. Wintraecken
{"title":"The reach of subsets of manifolds","authors":"J. Boissonnat, M. Wintraecken","doi":"10.1007/s41468-023-00116-x","DOIUrl":"https://doi.org/10.1007/s41468-023-00116-x","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42619158","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}
Pub Date : 2023-03-02DOI: 10.1007/s41468-023-00135-8
P. Gillespie
{"title":"Vietoris thickenings and complexes are weakly homotopy equivalent","authors":"P. Gillespie","doi":"10.1007/s41468-023-00135-8","DOIUrl":"https://doi.org/10.1007/s41468-023-00135-8","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42604295","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}
Pub Date : 2023-01-01Epub Date: 2022-10-05DOI: 10.1007/s41468-022-00102-9
Michał Lipiński, Jacek Kubica, Marian Mrozek, Thomas Wanner
We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in Mrozek (Found Comput Math 17(6):1585-1633, 2017). The generalization is threefold. First, we drop the restraining assumption in Mrozek (Found Comput Math 17(6):1585-1633, 2017) that every multivector must have a unique maximal element. Second, we define the dynamical system induced by the multivector field in a less restrictive way. Finally, we also change the setting from Lefschetz complexes to finite topological spaces. Formally, the new setting is more general, because every Lefschetz complex is a finite topological space, but the main reason for switching to finite topologcial spaces is because the latter better explain some peculiarities of combinatorial topological dynamics. We define isolated invariant sets, isolating neighborhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.
我们对Mrozek（Found Comput Math 17（6）：1585-16332017）中引入的组合多向量场的Conley Morse Forman理论进行了推广和推广。概括有三个方面。首先，我们放弃了Mrozek（Found Comput Math 17（6）:1585-16332017）中的约束假设，即每个多向量都必须具有唯一的极大元素。其次，我们以一种限制较少的方式定义了由多矢量场诱导的动力系统。最后，我们还将勒夫谢兹复形的设置改为有限拓扑空间。形式上，新的设置更一般，因为每个Lefschetz复形都是有限拓扑空间，但切换到有限拓扑空间的主要原因是后者更好地解释了组合拓扑动力学的一些特性。我们定义了孤立不变集、孤立邻域、康利指数和Morse分解。我们还建立了Conley指数和Morse不等式的可加性。
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