We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram...We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.展开更多
In the study of smooth functions on manifolds,min-max theory provides a mechanism for identifying critical values of a function.We introduce a discretized version of this theory associated to a discrete Morse function...In the study of smooth functions on manifolds,min-max theory provides a mechanism for identifying critical values of a function.We introduce a discretized version of this theory associated to a discrete Morse function on a(regular)cell complex.As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik-Schnirelmann theorem.展开更多
文摘We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.
文摘In the study of smooth functions on manifolds,min-max theory provides a mechanism for identifying critical values of a function.We introduce a discretized version of this theory associated to a discrete Morse function on a(regular)cell complex.As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik-Schnirelmann theorem.
