Article
Mathematics
John R. Klein, Lizhen Qin, Yang Su
Summary: We establish foundational results on Poincare accent spaces and apply them in various areas. Our applications include settling an old conjecture by C.T.C. Wall, showing the existence of a finite CW pair that satisfies relative Poincare accent duality in dimension n but fails to satisfy Poincare accent duality, and proving a relative version of a Gottlieb's result on Poincare accent duality and fibrations.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Astronomy & Astrophysics
Shinya Tomizawa, Ryotaku Suzuki
Summary: In this study, the inverse scattering method is applied to the static and biaxisymmetric Einstein equations to construct a nonrotating black lens inside a bubble of nothing. The horizon of this lens is topologically a lens space, L(n, 1) = S3/7Ln. The equilibrium of a static black lens is then discussed based on the force balance between expansion and gravitational attraction.
Article
Computer Science, Theory & Methods
Harshita Tiwari, Rekha Srivastava
Summary: The article presents the characterization of exponential objects in topological spaces by Escardo and Heckmann, and further extends it to Q-topological spaces, emphasizing that the proof method is not based on category theory.
FUZZY SETS AND SYSTEMS
(2021)
Article
Computer Science, Software Engineering
Yann-Situ Gazull, Alexandra Bac, Aldo Gonzalez-Lorenzo
Summary: Persistent homology is a method in algebraic topology that computes the homology of a growing object. It detects holes and provides information about their importance. This paper introduces two theoretical methods for computing hole measures in volumetric objects defined by surface meshes.
COMPUTER-AIDED DESIGN
(2023)
Article
Mathematics
Andreas Gross, Farbod Shokrieh
Summary: In this paper, we introduce a sheaf-theoretic approach to studying tropical homology. By establishing proper push-forwards and products, we show that tropical homology behaves similarly to classical Borel-Moore homology. We also define the tropical cycle class map and characterize the rational polyhedral spaces that satisfy Poincare-Verdier duality.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics
Jeremy Hahn, Dylan Wilson
Summary: The paper provides a new proof of the Segal conjecture for the cyclic group of order two, independent of Lin's theorem. The key component is a calculation of the Real topological Hochschild homology of F-2 as a Hopf algebroid. This calculation determines the E-2-page of the descent spectral sequence for the map NF2 -> F-2, leading to a new upper bound on the RO(C-2)-graded homotopy of NF2, which establishes the Segal conjecture as an immediate corollary.
ADVANCES IN MATHEMATICS
(2021)
Article
Computer Science, Artificial Intelligence
James R. Clough, Nicholas Byrne, Ilkay Oksuz, Veronika A. Zimmer, Julia A. Schnabel, Andrew P. King
Summary: This method introduces prior knowledge about the topology of segmented objects into the training process of neural networks for image or volume segmentation. The use of persistent homology allows for specifying desired topological features and driving the proposed segmentations to contain these features. The experiments demonstrate the effectiveness of embedding explicit prior knowledge in challenging segmentation tasks.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
(2022)
Article
Mathematics, Applied
E. Macias-Virgos, D. Mosquera-Lois, J. A. Vilches
Summary: The main goal of this paper is to prove a Poincare duality theorem in the context of finite posets, specifically focusing on the class of homologically bi-admissible finite posets.
TOPOLOGY AND ITS APPLICATIONS
(2023)
Article
Mathematics
Yunguang Yue, Xingwu Liu, Fengchun Lei, Jie Wu
Summary: In this work, we extend the topology-based framework and method for quantifying and classifying general resilient asynchronous complexity. We propose the arbitrary resilient asynchronous complexity theorem, which is applied to decision tasks in an iterated delayed model. We introduce two topological structures, delayed complex and reduced delayed complex, and build a topological computability model to investigate the properties of these structures and the computing power of the model. The derived theorem states that the time complexity of any arbitrary resilient asynchronous algorithm is proportional to the level of a reduced delayed complex.
Article
Mathematics
Cary Malkiewich, Kate Ponto
Summary: This article answers two conjectures made by Klein and Williams in the affirmative. Firstly, in a range of dimensions, the equivariant Reidemeister trace serves as a complete obstruction to removing n-periodic points from a self-map f. Secondly, this obstruction defines a class in topological restriction homology. These results are proven using duality and trace for bicategories, allowing for immediate generalizations, including a corresponding theorem for the fiberwise Reidemeister trace.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics, Applied
Sang-Eon Han
Summary: This article translates the importance of MW-topological spaces and proves that they satisfy the semi-T-3 separation axiom. By proposing several techniques, the article distinguishes between the semi-openness and semi-closedness of sets in MW-topological spaces and proves some important properties.
Article
Mathematics, Applied
A. A. El-Atik, Y. Tashkandy, S. Jafari, A. A. Nasef, W. Emam, M. Badr
Summary: This article focuses on studying DNA sequence mutations using multisets, relations, metric functions, topology, and association indices. Different methods of identifying mutations are utilized to aid biologists in decision-making.
Article
Mathematics, Applied
Yuxu Chen, Hui Kou
Summary: This paper investigates the property of core compactness in ordered topological spaces, with a particular focus on directed spaces. A series of characterizations of core compactness in directed spaces are presented. The results obtained in this paper are closely related to a long-standing open problem in the field of Topology.
Article
Mathematics, Applied
John R. Klein, Florian Naef
Summary: In this paper, we investigate the relationship between the total obstruction mu M, proposed by the first author, to the existence of a Poincaré embedding of the diagonal map M -> M xM in [17], and the Reidemeister trace of the identity map of M for a finitely dominated Poincaré duality space M. We then demonstrate that mu M vanishes when suitable conditions on the fundamental group of M are satisfied.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
Review
Mathematics, Applied
Eddy Kwessi
Summary: In this paper, the authors explore the use of topological tools to compare dimension reduction methods. They provide an overview of commonly used methods and topological notions, and compare these methods across persistent homologies. The results suggest that changes in homology landscapes could be a predictor of seizures.
Article
Mathematics
Tathagata Basak, Ryan Johnson
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Mathematics
Tathagata Basak
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2016)
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Mathematics
Daniel Allcock, Tathagata Basak
GEOMETRY & TOPOLOGY
(2016)
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Mathematics
Daniel Allcock, Tathagata Basak
GEOMETRY & TOPOLOGY
(2016)
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Mathematics
Tathagata Basak
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2012)
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Mathematics
Tathagata Basak
JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX
(2014)
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Mathematics
Tathagata Basak
JOURNAL OF ALGEBRA
(2007)
Article
Mathematics
Tathagata Basak
JOURNAL OF ALGEBRA
(2007)