Monotone networks

Abstract

Fundamental existence and uniqueness theorems for electrical networks of non-linear resistors are proved in an abstract form, as theorems of pure mathematics. The two groups from which the ‘currents’ and ‘voltage drops’ are drawn are permitted to be either the real numbers, or discrete subgroups of the reals. It is found that the uniqueness theory is deriv­able from extremum principles for certain convex functions associated with the networks, and that the existence theory is derivable from a single new theorem of graph theory. The abstract approach, besides revealing the logical structure of the subject more clearly than the ‘concrete’ approach, also (1) reveals the mathematical problem of solving a non­-linear network to be identical with certain extremum problems arising in non-electrical applications, (2) contributes a numerical method, since the constructions for the discrete case are algorithmic, and (3) permits the application of the theorems to problems of pure mathematics. Applications are not fully discussed; they will be treated at greater length in the appropriate technical journals.