Article
Multidisciplinary Sciences
Yu-Min Chung, Sarah Day, Chuan-Shen Hu
Summary: Mathematical morphology is a well-studied technique for image processing in various fields. This work introduces the concept of persistent homology to analyze morphological operations and extract information about topology and geometry in images. The authors demonstrate its effectiveness by developing an automated denoising approach and comparing it to state-of-the-art deep learning methods.
SCIENTIFIC REPORTS
(2022)
Article
Computer Science, Artificial Intelligence
Nicolas Boutry, Laurent Najman, Thierry Geraud
Summary: This paper examines the relationship between connected filters based on dynamics in mathematical morphology and persistence concept in persistent homology and Morse theory. It proves that they are equal on n-D Morse functions. This result paves the way for further study on the relationship between topological data analysis and mathematical morphology.
JOURNAL OF MATHEMATICAL IMAGING AND VISION
(2022)
Article
Computer Science, Software Engineering
Henry-Louis de Kergorlay, Ulrike Tillmann, Oliver Vipond
Summary: In this study, we investigate the homology of a random Cech-complex generated by a homogeneous Poisson process in a compact, unit volume, Riemannian manifold with boundary, denoted as M. Our main results provide two asymptotic threshold formulas, an upper threshold indicating the high probability of recovering the kth homology of M, and a lower threshold implying the improbability of the same. These thresholds have the same leading term and extend the previously established formulas for manifolds without boundary by Bobrowski-Weinberger and Bobrowski-Oliveira. However, the leading terms for the upper and lower thresholds differ when M has a boundary. Our analysis also identifies a specific type of homological cycle occurring near the boundary.
RANDOM STRUCTURES & ALGORITHMS
(2022)
Article
Mathematics, Applied
Slobodan Maletic, Miroslav Andjelkovic, Milan Rajkovic
Summary: The research focuses on characterizing potential mesoscale structures in complex networks, obtaining a simplicial complex from a mathematical graph to produce a clique complex. Higher-order organizational structures are naturally embedded in the hierarchical strata of a simplicial complex, and configurations are characterized using an observability parameter.
Article
Mathematics
Thomas J. X. Li, Christian M. Reidys
Summary: In this study, a topological framework of tau-structures was established to quantify evolutionary transitions between two RNA sequence-structure pairs. The loop complex of tau-structures captures intersections of loops in secondary structures, and the loop homology was computed, showing that only zeroth, first, and second homology groups are free. Additionally, the rank of the second homology group was found to equal the number of arc-components in a tau-structure, while the rank of the first homology group is given by a specific equation involving the Euler characteristic of the loop complex.
Article
Mathematics, Applied
Ryo Asai, Jay Shah
Summary: We introduce a new algorithm based on local homology for the structural analysis of finite abstract simplicial complexes. The algorithm computes a canonical stratification of a simplicial complex through an iterative and top-down procedure, attaching a stratified homotopy type to the complex. Using oo-categorical methods, we prove the correctness of the algorithm and provide a pseudocode implementation with polynomial time complexity. Experimental results confirm the linear running time of the algorithm.
JOURNAL OF PURE AND APPLIED ALGEBRA
(2022)
Article
Statistics & Probability
Omer Bobrowski
Summary: This study investigates the homology of random Cech complexes generated by a homogeneous Poisson process and examines the phenomenon of 'homological connectivity'. The results reveal sharp phase transitions describing homological connectivity in different dimensions and analyze the behavior of the complex within each critical window. The study also demonstrates the unique and simple shape of cycles obstructing homological connectivity and proves the convergence of the process counting the last obstructions to a Poisson process. Morse theory and its adaptation to distance functions are extensively used in the research to classify the critical points of random distance functions according to their effect on homology.
PROBABILITY THEORY AND RELATED FIELDS
(2022)
Article
Computer Science, Information Systems
Markus Wilhelm Jahn, Patrick Erik Bradley
Summary: This article calculates geometrically induced topology useful for applications such as simulations and navigation by presenting unpublished algorithms. It also analyzes the robustness of the implementations and presents the results. The findings show that most volumes can be extracted, but there are failures and various outcomes in computational geometry.
ISPRS INTERNATIONAL JOURNAL OF GEO-INFORMATION
(2022)
Article
Computer Science, Theory & Methods
Erin W. Chambers, Jeff Erickson, Kyle Fox, Amir Nayyeri
Summary: This article introduces algorithms for efficiently computing minimum (s, t)-cuts and global minimum cuts in undirected surface-embedded graphs. The algorithms can solve either problem in gO(g)n log log n or 2O(g)n log n time, depending on which is better. When g is a constant, the algorithms match the best running times known for computing minimum cuts in planar graphs. Efficient algorithms are also provided for finding a minimum-weight subgraph in a given Z2-homology class. The time complexity of these algorithms is (g + b)O(g+b)n log log n and 2O(g+b)n log n if G is embedded on a surface with genus g and b boundary components.
SIAM JOURNAL ON COMPUTING
(2023)
Article
Computer Science, Artificial Intelligence
Yule Vaz, Rodrigo Fernandes de Mello, Carlos Henrique Grossi Ferreira
Summary: The data clustering problem is crucial in machine learning, but there is limited theoretical framework literature with generalization guarantees. This manuscript introduces a new concept based on multi-dimensional persistent homology to analyze the conditions for clustering models to generalize data. The coarse-refinement dilemma highlights the need for a relaxation of Kleinberg's richness axiom to avoid unstable or unrepresentative partitions.
EXPERT SYSTEMS WITH APPLICATIONS
(2021)
Article
Astronomy & Astrophysics
Alfredo Giambrone, Adolfo Guarino, Emanuel Malek, Henning Samtleben, Colin Sterckx, Mario Trigiante
Summary: We provide the first holographic evidence for the existence of a nonsupersymmetric conformal manifold resulting from exactly marginal but supersymmetry-breaking deformations. Specifically, we construct a 2-parameter nonsupersymmetric deformation of a supersymmetric AdS nongeometric background in type 1IB string theory. We prove the perturbative stability of the nonsupersymmetric backgrounds and their protection against various nonperturbative instabilities. Additionally, we argue that diffeomorphism symmetry safeguards our solutions against higher-derivative string corrections.
Article
Computer Science, Information Systems
Markus Wilhelm Jahn, Patrick Erik Bradley
Summary: This study develops a topological access method (TOAM) based on the concept of Property Graph to perform topological queries on geographic data. The experimental test shows that the Euler characteristic is helpful for data validation. This research is important for future innovative applications.
ISPRS INTERNATIONAL JOURNAL OF GEO-INFORMATION
(2022)
Article
Physics, Fluids & Plasmas
Leonie Neuhaeuser, Renaud Lambiotte, Michael T. Schaub
Summary: The study indicates that in network systems with time-dependent, multiway interactions, the convergence speed of consensus dynamics is slower than systems with only pairwise interactions, and slower than consensus dynamics on corresponding static networks. Additionally, the final consensus value in a temporal system may differ significantly from the consensus value on an aggregated, static hypergraph, with early movers having a greater influence.
Article
Mathematics
Kouyemon Iriye, Daisuke Kishimoto
Summary: The notions of Golodness and tightness, derived from algebra and geometry, respectively, are proven to be equivalent for 3-manifold triangulations by topologically characterizing a polyhedral product for a tight-neighborly manifold triangulation of dimension > 3.
ALGEBRAIC AND GEOMETRIC TOPOLOGY
(2023)
Article
Mathematics
Richard H. Bamler, Bruce Kleiner
Summary: In this paper, we investigate the relationship between the diffeomorphism group and the isometry group of a compact orientable non-Haken 3-manifold X modeled on the Thurston geometry Nil. By combining previous work, we determine the homotopy type of Diff(X) for any compact orientable prime 3-manifold X.
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2023)
