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A maximum principle for node voltages in finitely structured, transfinite, electrical networks

Abstract

Transfinite electrical networks have unique finite-powered voltage-current regimes given in terms of branch voltages and branch currents, but they do not in general possess unique node voltages. However, if their structures are sufficiently restricted, those node voltages will exist and will satisfy a maximum principle much like that which holds for ordinary infinite electrical networks. The structure that is imposed in order to establish these results generalized the idea of local-finiteness. Other properties that do not hold in general for transfinite networks but do hold under the imposed structure are Kirchhoff's current laws for nodes of any ranks and the permissibility of connecting pure voltage sources to such nodes. This work lays the foundation for a theory of transfinite random walks, which will be the subject of a subsequent work.

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Additional information

This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.

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Zemanian, A.H. A maximum principle for node voltages in finitely structured, transfinite, electrical networks. Potential Anal 5, 73–101 (1996). https://doi.org/10.1007/BF00276698

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Mathematics Subject Classification (1991)

  • Primary 05C99, 31C45
  • Secondary 94C05

Key words

  • Transfinite graphs
  • transfinite electrical networks
  • maximum principle
  • node voltages