Ukrainian Mathematical Journal

, Volume 65, Issue 7, pp 1087–1110 | Cite as

Trees as Level Sets for Pseudoharmonic Functions in the Plane

  • E. O. Polulyakh

Let T be a finite or infinite tree and let V 0 be the set of all vertices of T of valency 1. We propose a sufficient condition for the image of the imbedding ψ: T \V 0\( {{\mathbb{R}}^2} \) to be a level set of a pseudoharmonic function.


Open Subset Open Neighborhood Edge Incident Jordan Curve Plane Mapping 
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  1. 1.
    M. Morse, Topological Methods in the Theory of Functions of a Complex Variable, Princeton University Press, Princeton (1947).zbMATHGoogle Scholar
  2. 2.
    E. Polulyakh and I. Yurchuk, On the Pseudo-Harmonic Functions Defined on a Disk, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2009).Google Scholar
  3. 3.
    V. A. Rokhlin and D. B. Fuks, A First Course in Topology. Geometric Chapters [in Russian], Nauka, Moscow (1977).Google Scholar
  4. 4.
    M. H. A. Newman, Elements of the Topology of Plane Sets of Points, Cambridge Univ. Press, Cambridge (1964).Google Scholar
  5. 5.
    C. Berge, Théorie des Graphes et Ses Applications, Dunod, Paris (1958).Google Scholar
  6. 6.
    O. Ore, Theory of Graphs, American Mathematical Society, Providence, RI (1962).Google Scholar
  7. 7.
    Y. Tôki, “A topological characterization of pseudo-harmonic functions,” Osaka Math. J., 3, No. 1, 101–122 (1951).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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