Logo image
Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams
Journal article   Peer reviewed

Monopoles, Dipoles, and Harmonic Functions on Bratteli Diagrams

Sergey Bezuglyi and Palle Jorgensen
Acta Applicandae Mathematicae, Vol.159(1), pp.169-224
02/15/2019
DOI: 10.1007/s10440-018-0189-7
url
https://arxiv.org/pdf/1508.01253View
Open Access

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.
Mathematics Partial Differential Equations Laplace operator Computational Mathematics and Numerical Analysis Green’s function Random walk Probability Theory and Stochastic Processes Bratteli diagram 37L30 Harmonic function Monopole 47L50 Electrical network Pascal graph Calculus of Variations and Optimal Control; Optimization Semibranching function system 37B10 Dipole Applications of Mathematics 60J45 Symmetry

Details

Metrics

Logo image