A characterization theorem for cellular maps

Haver, William E.

doi:10.1090/s0002-9904-1970-12638-6

Cited by 7 publications

(4 citation statements)

References 7 publications

(1 reference statement)

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“…The first question is at the heart of topology, as it concerns approximation of a mapping with homeomorphisms. We recall that limits of homeomorphisms are cellular mappings ( [23] and [24]), meaning that the inverse image of a point is an intersection of a decreasing sequence of n-cells, the notion introduced by Morton Brown [13]. In fact, Armentrout showed [3] that cellular mappings of an n-manifold onto itself can be approximated with homeomorphisms; see also Siebenmann [58].…”

Section: Lemma 91 (Equicontinuity) the Family Ementioning

confidence: 99%

Deformations of finite conformal energy: Boundary behavior and limit theorems

Iwaniec

Onninen

2011

Trans. Amer. Math. Soc.

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Abstract. We study homeomorphisms h : X onto −→ Y between two bounded domains in R n having finite conformal energyWe consider the behavior of such mappings, including continuous extension to the closure of X and injectivity of h : X → Y. In general, passing to the weak W 1,n -limit of a sequence of homeomorphisms h j : X → Y one loses injectivity. However, if the mappings in question have uniformly bounded L 1 -average of the inner distortion, then, for sufficiently regular domains X and Y, their limit map h : X onto −→ Y is a homeomorphism. Moreover, the inverse map−→ X enjoys finite conformal energy and has integrable inner distortion as well.

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Section: Lemma 91 (Equicontinuity) the Family Ementioning

confidence: 99%

Deformations of finite conformal energy: Boundary behavior and limit theorems

Iwaniec

Onninen

2011

Trans. Amer. Math. Soc.

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Section: Remarks On the Existence Of A Minimizing Mapmentioning

confidence: 99%

Untitled

2011

Trans. Amer. Math. Soc.

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We consider the behavior of such mappings, including continuous extension to the closure of X and injectivity of h : X → Y. In general, passing to the weak W 1,n-limit of a sequence of homeomorphisms h j : X → Y one loses injectivity. However, if the mappings in question have uniformly bounded L 1-average of the inner distortion, then, for sufficiently regular domains X and Y, their limit map h : X onto −→ Y is a homeomorphism. Moreover, the inverse map f = h −1 : Y onto −→ X enjoys finite conformal energy and has integrable inner distortion as well.

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“…The conditions on f ensure that f is a cellular map, and therefore it can be approximated by homeomorphisms -for the definition of these terms, and the justification of the last statement, see [9] and (for the dimension 4 case) [14]. The existence of an extension F with the properties promised by Proposition A.1 now follows by Lemma 1 of [9]. Now, given topological spaces Y, Z, and a map f : Y → Z, let R f be the equivalence relation on Y induced by f , i.e.…”

Section: Proofmentioning

confidence: 99%

A regular decomposition of the edge-product space of phylogenetic trees

Gill

Linusson

Moulton

et al. 2008

Advances in Applied Mathematics

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We investigate the topology and combinatorics of a topological space called the edge-product space that is generated by the set of edge-weighted finite labelled trees. This space arises by multiplying the weights of edges on paths in trees, and is closely connected to tree-indexed Markov processes in molecular evolutionary biology. In particular, by considering combinatorial properties of the Tuffley poset of labelled forests, we show that the edge-product space has a regular cell decomposition with face poset equal to the Tuffley poset.Original Publication:Jonna Gill, Svante Linusson, Vincent Moulton and Mike Steel, A regular decomposition of the edge-product space of phylogenetic trees, 2008, Advances in Applied Mathematics, (14), 2, 158-176.http://dx.doi.org/10.1016/j.aam.2006.07.007Copyright: Elsevier Science B.V., Amsterdamhttp://www.elsevier.com

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A characterization theorem for cellular maps

Cited by 7 publications

References 7 publications

Deformations of finite conformal energy: Boundary behavior and limit theorems

Deformations of finite conformal energy: Boundary behavior and limit theorems

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A regular decomposition of the edge-product space of phylogenetic trees

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