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Polytope Novikov homology.
J Fixed Point Theory Appl
September 2021
Department of Mathematics, ETH Zürich, Zurich, Switzerland.
Let be a closed manifold and a polytope. For each , we define a Novikov chain complex with a multiple finiteness condition encoded by the polytope . The resulting polytope Novikov homology generalizes the ordinary Novikov homology.
View Article and Find Full Text PDFKhovanov homotopy type, periodic links and localizations.
Math Ann
February 2021
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland.
Given an -periodic link , we show that the Khovanov spectrum constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of to the equivariant Khovanov homology of constructed by the second author. The action of Steenrod algebra on the cohomology of gives an extra structure of the periodic link.
View Article and Find Full Text PDFEvaluating State Space Discovery by Persistent Cohomology in the Spatial Representation System.
Front Comput Neurosci
April 2021
Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA, United States.
Persistent cohomology is a powerful technique for discovering topological structure in data. Strategies for its use in neuroscience are still undergoing development. We comprehensively and rigorously assess its performance in simulated neural recordings of the brain's spatial representation system.
View Article and Find Full Text PDFHypergraph-based persistent cohomology (HPC) for molecular representations in drug design.
Brief Bioinform
September 2021
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore.
Artificial intelligence (AI) based drug design has demonstrated great potential to fundamentally change the pharmaceutical industries. Currently, a key issue in AI-based drug design is efficient transferable molecular descriptors or fingerprints. Here, we present hypergraph-based molecular topological representation, hypergraph-based (weighted) persistent cohomology (HPC/HWPC) and HPC/HWPC-based molecular fingerprints for machine learning models in drug design.
View Article and Find Full Text PDFPersistent Cohomology for Data With Multicomponent Heterogeneous Information.
SIAM J Math Data Sci
May 2020
Department of Mathematics, Department of Biochemistry and Molecular Biology, Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824.
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and reveal certain interaction patterns in terms of topological invariants. However, in addition to the geometric information, there is a wide variety of nongeometric information of molecular structures, such as element types, atomic partial charges, atomic pairwise interactions, and electrostatic potential functions, that is not described by persistent homology.
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