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Ukrainian Mathematical Journal

, Volume 67, Issue 10, pp 1572–1583 | Cite as

Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. II

  • E. A. Polulyakh
Article
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We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema and if a point is not a local extremum, then there exist its neighborhood and a number n ∈ ℕ such that the function restricted to this neighborhood is topologically conjugate to Re z n in a certain neighborhood of zero. Given f : M 2 → ℝ, let Γ K−R (f) be a quotient space of M 2 with respect to its partition formed by the components of level sets of the function f. It is known that the space Γ K−R (f) is a topological graph if M 2 is compact. In the first part of the paper, we introduced the notion of graph with stalks that generalizes the notion of topological graph. For noncompact M 2 , we present three conditions sufficient for Γ K−R (f) to be a graph with stalks. In the second part, we prove that these conditions are also necessary in the case M 2 = ℝ2 . In the general case, one of our conditions is not necessary. We provide an appropriate example.

Keywords

Singular Point Regular Point Local Extremum Hausdorff Space Topological Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    E. A. Polulyakh, “Kronrod–Reeb graphs of functions on noncompact two-dimensional surfaces. IUkr. Mat. Zh., 67, No. 3, 375–396 (2015); English translation: Ukr. Math. J., 67, No. 3, 431–454 (2015).Google Scholar
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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • E. A. Polulyakh
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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