Using methods from Convex Analysis, for each generalized pressure function we define an upper semi-continuous affine entropy-like map, establish abstract variational principle both countably and finitely additive probability measures prove that equilibrium states always exist. We show this conceptual approach imparts a new insight on dynamical systems without measure with maximal entropy, may b...

We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem arises in a number of other contexts, e.g. it is a relaxation of general mixed-integer programming. We show how strong inequalities valid for the semi-continuous knapsack ...