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Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

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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.

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  1. Sergey Bezuglyi

    Present address: Department of Mathematics, University of Iowa, Iowa City, 52242, IA, USA

Authors and Affiliations

  1. Department of Mathematics, Institute for Low Temperature Physics, Kharkiv, 61103, Ukraine

    Sergey Bezuglyi

  2. Department of Mathematics, University of Iowa, Iowa City, 52242, IA, USA

    Palle E. T. Jorgensen

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  1. Sergey Bezuglyi
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  2. Palle E. T. Jorgensen
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Correspondence to Sergey Bezuglyi.

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To the memory of Ola Bratteli.

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Bezuglyi, S., Jorgensen, P.E.T. Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams. Acta Appl Math 159, 169–224 (2019). https://doi.org/10.1007/s10440-018-0189-7

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