Lecture Notes of the Unione Matematica Italiana

Harmonic Functions and Potentials on Finite or Infinite Networks

Authors: Anandam, Victor

  • 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.
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About this book

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.

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)


Table of contents (5 chapters)

  • Laplace Operators on Networks and Trees

    Anandam, Victor

    Pages 1-20

  • Potential Theory on Finite Networks

    Anandam, Victor

    Pages 21-44

  • Harmonic Function Theory on Infinite Networks

    Anandam, Victor

    Pages 45-90

  • Schrödinger Operators and Subordinate Structures on Infinite Networks

    Anandam, Victor

    Pages 91-108

  • Polyharmonic Functions on Trees

    Anandam, Victor

    Pages 109-132

Buy this book

eBook $34.99
price for USA (gross)
  • ISBN 978-3-642-21399-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.95
price for USA
  • ISBN 978-3-642-21398-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Harmonic Functions and Potentials on Finite or Infinite Networks
Authors
Series Title
Lecture Notes of the Unione Matematica Italiana
Series Volume
12
Copyright
2011
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-21399-1
DOI
10.1007/978-3-642-21399-1
Softcover ISBN
978-3-642-21398-4
Series ISSN
1862-9113
Edition Number
1
Number of Pages
X, 141
Topics