Network topology: Matrices associated with graphs, the short circuit and open circuit operations, their generalization through the use of ideal transformers and vector space operations corresponding to these operations; Theorems of Tellegen and Minty: Formal equivalence, areas of applications; The Implicit Duality Theorem and its applications: Multiport decomposition, ideal transformer resulting from the connection of ideal transformers, adjoint networks and systems, networks with decomposition methods based on altering network topology, ideal diode, ideal transformer, resistor circuits and their relation to Linear and Quadratic Programming.