A MAXIMUM PRINCIPLE FOR NODE VOLTAGES IN FINITELY STRUCTURED, TRANSFINITE, ELECTRICAL NETWORKS

Transfinite electrical networks have unique finite-powered voltage-cur rent regimes given in terms of branch voltages and branch currents, bu t they do not in general possess unique node voltages. However, if the ir structures are sufficiently restricted, those node voltages will ex ist and will satisfy a maximum principle much like that which holds fo r ordinary infinite electrical networks. The structure that is imposed in order to establish these results generalizes the idea of local-fin iteness, Other properties that do not hold in general for transfinite networks but do hold under the imposed structure are Kirchhoff's curre nt laws for nodes of any ranks and the permissibility of connecting pu re voltage sources to such nodes. This work lays the foundation for a theory of transfinite random walks, which will be the subject of a sub sequent work.