The number of excellent discrete Morse functions on graphs
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Discrete Morse theory
Discrete Morse theory
Morse potential
Circle-valued Morse theory
Manifold (fluid mechanics)
Morse homology
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We introduce an algorithm for construction of the Morse hierarchy, i.e., a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse decompositions, which can be used to obtain Morse sets of user-prescribed stability. More stable Morse decompositions are coarser, i.e., they consist of larger Morse sets. In this work, we develop an algorithm for tracking the growth of Morse sets and topological events (mergers) that they undergo as their stability is gradually increased. The resulting Morse hierarchy can be explored interactively. We provide examples demonstrating that it can provide a useful coarse overview of the vector field topology.
Discrete Morse theory
Morse potential
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Discrete Morse theory
Morse potential
Link (geometry)
Circle-valued Morse theory
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Motivated by the study of the recurrent orbits in a Morse set of a Morse decomposition, we introduce the concept of Morse predecomposition of an isolated invariant set in the setting of combinatorial and classical dynamical systems. We prove that a Morse predecomposition indexed by a poset is a Morse decomposition and we show how a Morse predecomposition may be condensed back to a Morse decomposition.
Discrete Morse theory
Circle-valued Morse theory
Morse potential
Handlebody
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The concepts of Morse decompositions and dynamic Morse decompositions are equivalent for flows. In this paper we show that these concepts are not equivalent for Morse decompositions of semigroups of homeomorphisms on topological spaces.We give an example of a dynamic Morse decompositions on compactifications of topological spaces. Other examples of Morse decompositions are also provided.
Discrete Morse theory
Circle-valued Morse theory
Morse potential
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We prove an infinite analogue of the main theorem of discrete Morse theory formulated in terms of discrete Morse matchings. Our theorem holds under the assumption that the given Morse matching induces finitely many equivalence classes of infinite directed simple paths. A homological version of the theorem is also given.
Discrete Morse theory
Circle-valued Morse theory
Handlebody
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Discrete Morse theory
Circle-valued Morse theory
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From the topological viewpoint, Morse shellings of finite simplicial complexes are {\it pinched} handle decompositions and extend the classical shellings. We prove that every discrete Morse function on a finite simplicial complex induces Morse shellings on its second barycentric subdivision whose critical tiles-or pinched handles-are in oneto-one correspondence with the critical faces of the function, preserving the index. The same holds true, given any smooth Morse function on a closed manifold, for any piecewise-linear triangulation on it after sufficiently many barycentric subdivisions.
Barycentric coordinate system
Discrete Morse theory
Manifold (fluid mechanics)
Circle-valued Morse theory
Piecewise linear manifold
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Discrete Morse theory
Representation
Circle-valued Morse theory
Morse potential
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