Harmonic Functions and Potentials on Finite or Infinite Networks

  • Victor Anandam
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 12)

Table of contents

  1. Front Matter
    Pages i-x
  2. Victor Anandam
    Pages 1-20
  3. Victor Anandam
    Pages 21-44
  4. Victor Anandam
    Pages 109-132
  5. Back Matter
    Pages 133-141

About this book

Introduction

Random walks, Markov chains and electrical networks serve as an introduction to the study of real-valued functions on finite or infinite graphs, with appropriate interpretations using probability theory and current-voltage laws. The relation between this type of function theory and the (Newton) potential theory on the Euclidean spaces is well-established. The latter theory has been variously generalized, one example being the axiomatic potential theory on locally compact spaces developed by Brelot, with later ramifications from Bauer, Constantinescu and Cornea. A network is a graph with edge-weights that need not be symmetric. This book presents an autonomous theory of harmonic functions and potentials defined on a finite or infinite network, on the lines of axiomatic potential theory. Random walks and electrical networks are important sources for the advancement of the theory.

Keywords

31C20; 31D05; 30F20; 31A30; 15A09 Discrete Laplace and Schrödinger operators Discrete harmonic functions and potentials Flux in parabolic networks Ploypotentials in hyperbolic networks Subordinate harmonic structures

Authors and affiliations

  • Victor Anandam
    • 1
  1. 1.Ramanujan Inst. for Advanced Study in Ma, Department of MathematicsUniversity of MadrasChennaiIndia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-21399-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2011
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-21398-4
  • Online ISBN 978-3-642-21399-1
  • Series Print ISSN 1862-9113