Hyperreal operating points for nonlinear transfinite networks

A transfinite nonlinear resistive network need not have any operating point in the sense that Kichhoff's current law may be violated at some nodes of infinite degree and Kirchhoff's voltage law may be violated around some transfinite loops. This is so when standard analysis is used to determine branch voltages and currents. However, by borrowing some techniques of nonstandard analysis, we can show that Kirchhoff's laws will be satisfied everywhere when branch voltages and currents are determined as hyperreals (i.e., as nonstandard numbers).