Maximum principles for node voltages and branch currents in transfinite resistive networks

The classical maximum um principle for finite linear resistive networks asserts that every node voltage in a sourceless subnetwork is no greater (resp. no less) than the maximum (resp. minimum) node voltage at the boundary nodes of the subnetwork. A related result is that the absolute value of every branch current in the sourceless subnetwork is no greater than the sum of the absolute values of all the currents in all branch cuts within the subnetwork at the boundary nodes. These principles are extended to transfinite networks. Their proofs are far more complicated than those for the classical case. This is a consequence of the difficulty that Kirchhoff´s laws are not always satisfied in transfinite networks. (Tellegen´s equation is used instead.)