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
Acta Applicandae Mathematicae – Springer Journals
Published: May 31, 2018
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