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© 2011

Harmonic Functions and Potentials on Finite or Infinite Networks

Benefits

  • Number of examples to illustrate the main theory.

  • Historical perspectives included to show the development of potential theory in various forms.

  • Self-contained text for an easy reading.

Book

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

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

Bibliographic information

Reviews

From the reviews:

“In this book a potential-theoretic style of the theory is built into the framework of finite or infinite networks. The motivation of the book is to build a function theory on networks reflecting ideas of potential theory on locally compact spaces. … The book is written in a reader-friendly way and contains various potential-theoretic results … .” (Sirkka-Liisa Eriksson, Zentralblatt MATH, Vol. 1239, 2012)

“The book under review is a treatise of the potential theory on a network, that is, a graph with edge weights that need not be symmetric. … Besides being a very useful resource on the current important developments of the subject, this book has the potential even to change the mindset of those who are vocal critics of axiomatic potential theory, which is viewed by some as an abstruse and unappealing field.” (Flavia Colonna, Mathematical Reviews, Issue 2012 h)