Pub Date : 2024-01-01Epub Date: 2024-09-03DOI: 10.1007/s41468-024-00189-2
Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis
Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors , , that assign, to each metric pair (X, A), a pointed metric space . Moreover, we show that is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that preserves several useful metric properties, such as completeness and separability, for , and geodesicity and non-negative curvature in the sense of Alexandrov, for . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on , , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, , and , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.
持久图是拓扑数据分析中的核心对象。在本文中，我们将研究持久图空间的局部和全局几何特性。为此，我们构建了一系列函数 D p , 1 ≤ p ≤ ∞ , 为每个度量对 (X, A) 分配一个尖度量空间 D p ( X , A ) 。此外，我们证明 D ∞ 在度量对的格罗莫夫-豪斯多夫收敛性方面是连续的，并证明 D p 保留了几个有用的度量特性，如对于 p∈ [ 1 , ∞ ) 的完备性和可分性，以及大地性和非大地性。 的情况下，D p 保留了几个有用的度量特性，如完整性和可分性；在 p = 2 的情况下，D p 保留了大地性和亚历山德罗夫意义上的非负曲率。对于后一种情况，我们描述了空图处方向空间的度量。我们还证明了在 D p ( X , A ) 上的博尔概率度量的弗雷谢特均值集，1 ≤ p ≤ ∞，具有有限第二矩和紧凑支持，是非空的。作为几何框架的一个应用，我们证明了欧氏持久图空间 D p ( R 2 n , Δ n ) , 1 ≤ n 且 1 ≤ p ∞ 具有无限覆盖维、豪斯多夫维、渐近维、阿苏阿德维和阿苏阿德-纳加塔维。
{"title":"Metric geometry of spaces of persistence diagrams.","authors":"Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis","doi":"10.1007/s41468-024-00189-2","DOIUrl":"https://doi.org/10.1007/s41468-024-00189-2","url":null,"abstract":"<p><p>Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors <math><msub><mi>D</mi> <mi>p</mi></msub> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi></mrow> </math> , that assign, to each metric pair (<i>X</i>, <i>A</i>), a pointed metric space <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> . Moreover, we show that <math><msub><mi>D</mi> <mi>∞</mi></msub> </math> is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that <math><msub><mi>D</mi> <mi>p</mi></msub> </math> preserves several useful metric properties, such as completeness and separability, for <math><mrow><mi>p</mi> <mo>∈</mo> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo></mrow> </math> , and geodesicity and non-negative curvature in the sense of Alexandrov, for <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn></mrow> </math> . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo></mrow> </mrow> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi></mrow> </math> , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, <math> <mrow><msub><mi>D</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <msup><mrow><mi>R</mi></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>,</mo> <msub><mi>Δ</mi> <mi>n</mi></msub> <mo>)</mo></mrow> </mrow> </math> , <math><mrow><mn>1</mn> <mo>≤</mo> <mi>n</mi></mrow> </math> and <math><mrow><mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.</p>","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 8","pages":"2197-2246"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11541355/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142634046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.1007/s41468-023-00151-8
Susumu Nishimura
{"title":"Defining logical obstruction with fixpoints in epistemic logic","authors":"Susumu Nishimura","doi":"10.1007/s41468-023-00151-8","DOIUrl":"https://doi.org/10.1007/s41468-023-00151-8","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"79 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134900728","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 : 2023-11-13DOI: 10.1007/s41468-023-00147-4
Yicheng Fei, Xaq Pitkow
Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.
{"title":"Generalization of graph network inferences in higher-order graphical models","authors":"Yicheng Fei, Xaq Pitkow","doi":"10.1007/s41468-023-00147-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00147-4","url":null,"abstract":"Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"61 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283777","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 : 2023-11-13DOI: 10.1007/s41468-023-00150-9
Pierre Fraigniaud, Ran Gelles, Zvi Lotker
Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.
{"title":"The topology of randomized symmetry-breaking distributed computing","authors":"Pierre Fraigniaud, Ran Gelles, Zvi Lotker","doi":"10.1007/s41468-023-00150-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00150-9","url":null,"abstract":"Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"9 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993015","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 : 2023-11-10DOI: 10.1007/s41468-023-00148-3
Ron Rosenthal
{"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}
Pub Date : 2023-11-04DOI: 10.1007/s41468-023-00149-2
Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela
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.
受Mrozek等人(common Nonlinear Sci numerical Simul 108:106 - 226,2022)的理论结果的启发，我们提出了一种平面ODE的横向元胞分解算法。然后，我们使用相关的组合多向量场算法检测周期轨道的Conley指数的孤立不变量集的存在性，并允许一个组合poincarcarr截面。这种结构与Mrozek等人(2022)的理论结果相结合，产生了一种自动化计算机辅助证明ODE周期解存在性的方法。
{"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}
Pub Date : 2023-10-28DOI: 10.1007/s41468-023-00139-4
Bruno Benedetti, Crystal Lai, Davide Lofano, Frank H. Lutz
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 $$dle 6$$ d≤6 , 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, $$(14k+1)$$ (14k+1) -vertex triangulations of a new series of Bing’s houses with k rooms, $$kge 3$$ k≥3 , which all can be deformed to a point using only six pure elementary expansions.
摘要:我们实现了一个RSHT(随机简单同伦)算法来研究简单复合体的简单同伦类型，特别关注高维复合体的可收缩空间和寻找子结构。该算法结合了初等简单坍缩和纯初等展开。对于$$dle 6$$ d≤6的三角化d流形，我们证明RSHT减少为(随机)双星翻转。在我们测试RSHT的许多例子中，我们描述了Abalone的显式15顶点三角剖分，更一般地说，$$(14k+1)$$ (14 k + 1)顶点三角剖分，包含k个房间的新系列Bing的房子，$$kge 3$$ k≥3，它们都可以变形为一个点，仅使用六个纯初等展开。
{"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}
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.
{"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}
Pub Date : 2023-10-21DOI: 10.1007/s41468-023-00146-5
Tadas Temčinas, Vidit Nanda, Gesine Reinert
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.
{"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}
Pub Date : 2023-10-21DOI: 10.1007/s41468-023-00141-w
Nikolas C. Schonsheck, Stefan C. Schonsheck
{"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}
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