Formulae on first and second derivatives of various functions associated with a linear nullator–norator–resistance network such as total input power, driving-point and transfer resistances with respect to parameters are established. As a consequence, the concavity of the driving-point resistance with respect to the system of parameters is obtained which generalizes a scalar result of Schneider. An example is given showing that the driving-point resistance R of a nonreciprocal one-port is not monotone or convex or concave with respect to the system of resistances which shows that the Cohn–Vratsanos and the Shannon–Hagelbarger theorems which characterize R of reciprocal one-port cannot be extended in this way. Next, a simplified variant of the Shannon–Hagelbarger theorem is used to derive separate necessary and sufficient conditions characterizing always well-posed, sometimes ill-posed and always ill-posed classes of linear resistive circuit structures introduced and characterized by Hasler, both new in formulation and proof. This reveals that the form of the second partial derivative of the resistance function is responsible for various kinds of the structural solvability of linear circuits. Alternative "if and only if" criteria for these classes are established. They involve replacements of reciprocal circuit elements by combinations of contractions and removals leading to pairs of complementary directed nullator and directed norator trees with appropriately defined signs, and resemble therefore earlier famous Willson–Nielsen feedback structure and Chua–Nishi cactus graph criteria for circuits containing traditional controlled sources. Finally, the qualitative parts of the Cohn–Vratsanos and the Shannon–Hagelbarger theorems are shown to be simple consequences of much more general principles governing all aspects of life, such as maximal entropy and energy conservation laws.
Real polynomial diffeomorphisms with maximal entropy: II. Small Jacobian