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Nontrivial solutions of discrete Kirchhoff-type problems via Morse theory
Advances in Nonlinear Analysis, 2022 In this article, we study discrete Kirchhoff-type problems when the nonlinearity is resonant at both zero and infinity. We establish a series of results on the existence of nontrivial solutions by combining variational method with Morse theory.
Long Yuhua
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Discrete Stratified Morse Theory
Discrete and Computational Geometry, 2022 Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson. We describe the basics of this theory and prove fundamental theorems relating the topology of a general ...
Kevin P Knudson, Bei Wang
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Toward Optimality in Discrete Morse Theory [PDF]
Experimental Mathematics, 2003 Morse theory is a fundamental tool for investigating the topology of smooth manifolds. This tool has been extended to discrete structures by Forman, which allows combinatorial analysis and direct computation. This theory relies on discrete gradient vector fields, whose critical elements describe the topology of the structure.
Thomas Lewiner +2 more
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Equivariant discrete Morse theory
Discrete Mathematics, 2009 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Discrete Morse theory on digraphs [PDF]
Pure and Applied Mathematics Quarterly, 2021 In this paper, we give a necessary and sufficient condition that discrete Morse functions on a digraph can be extended to be Morse functions on its transitive closure, from this we can extend the Morse theory to digraphs by using quasi-isomorphism between path complex and discrete Morse complex, we also prove a general sufficient condition for digraphs
Lin, Yong, Wang, Chong, Yau, Shing-Tung
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The Elser nuclei sum revisited [PDF]
Discrete Mathematics & Theoretical Computer Science, 2021 Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ pandemic if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a ...
Darij Grinberg
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Parameterized Complexity of Discrete Morse Theory [PDF]
ACM Transactions on Mathematical Software, 2013 Optimal Morse matchings reveal essential structures of cell complexes that lead to powerful tools to study discrete geometrical objects, in particular, discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds through a reduction to the erasability problem.
Benjamin A. Burton +3 more
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Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems
Advances in Difference Equations, 2007 An existence result of multiple periodic solutions to the asymptotically linear discrete Hamiltonian systems is obtained by using the Morse index theory.
Bo Zheng
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Merging Discrete Morse Vector Fields: A Case of Stubborn Geometric Parallelization
Algorithms, 2021 We address the basic question in discrete Morse theory of combining discrete gradient fields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm
Douglas Lenseth, Boris Goldfarb
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Combinatorial Topology of Toric arrangements [PDF]
Discrete Mathematics & Theoretical Computer Science, 2013 We prove that the complement of a complexified toric arrangement has the homotopy type of a minimal CW-complex, and thus its homology is torsion-free.
Giacomo d'Antonio, Emanuele Delucchi
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