# Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams Acta Appl Math https://doi.org/10.1007/s10440-018-0189-7 Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams 1,2 3 Sergey Bezuglyi · Palle E.T. Jorgensen Received: 2 August 2016 / Accepted: 16 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract In our study of electrical networks we develop two themes: ﬁnding explicit for- mulas 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 prop- erties 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 (station- ary, 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 1 Introduction The interest in discrete harmonic analysis includes both classical roots, as well as new and recent research http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

# Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

Acta Applicandae Mathematicae, Volume OnlineFirst – May 31, 2018
56 pages

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
D.O.I.
10.1007/s10440-018-0189-7
Publisher site
See Article on Publisher Site

### Abstract

Acta Appl Math https://doi.org/10.1007/s10440-018-0189-7 Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams 1,2 3 Sergey Bezuglyi · Palle E.T. Jorgensen Received: 2 August 2016 / Accepted: 16 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract In our study of electrical networks we develop two themes: ﬁnding explicit for- mulas 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 prop- erties 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 (station- ary, 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 1 Introduction The interest in discrete harmonic analysis includes both classical roots, as well as new and recent research

### Journal

Acta Applicandae MathematicaeSpringer Journals

Published: May 31, 2018

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