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Acta Applicandae Mathematicae

, Volume 159, Issue 1, pp 169–224 | Cite as

Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

  • Sergey BezuglyiEmail author
  • Palle E. T. Jorgensen
Article
  • 25 Downloads

Abstract

In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy.

Keywords

Bratteli diagram Laplace operator Random walk Electrical network Monopole Dipole Harmonic function Semibranching function system Pascal graph Green’s function Symmetry 

Mathematics Subject Classification

37B10 37L30 47L50 60J45 

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Authors and Affiliations

  1. 1.Department of MathematicsInstitute for Low Temperature PhysicsKharkivUkraine
  2. 2.Department of MathematicsUniversity of IowaIowa CityUSA
  3. 3.Department of MathematicsUniversity of IowaIowa CityUSA

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