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Mathematics - Analysis | Harmonic Functions and Potentials on Finite or Infinite Networks

Harmonic Functions and Potentials on Finite or Infinite Networks

Anandam, Victor

2011, X, 141p.

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  • 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.

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.

Content Level » Research

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

Related subjects » Analysis - Dynamical Systems & Differential Equations

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