... (M) and ᏺ(M) for the range space and the null space of a matrix M, respectively, spec(M) for the spectrum of the square matrix M, rank(M) for the rank of the matrix M, ind(M) for the index of ... statement of the first law of thermodynamics for the discrete-time large-scale dynamical system Ᏻ and gives a precise formulation of the equivalence between variation in system internal energy and heat ... (3.15) for any k2 ≥ k1 ≥ k0 and S(·) ∈ ᐁ, is called the entropy of Ᏻ Next, we show that (3.13) guarantees the existence of an entropy function for Ᏻ For this result, define, the available entropy of...