In this paper, the continuous optimal control theory is used to model and solve the maximum entropy problem for a continuous random variable. The maximum entropy principle provides a method to obtain least-biased probability density function (Pdf) estimation. In this paper, to find a closed form solution for the maximum entropy problem with any number of moment constraints, the entropy is consi...

Part 1. Analogies 3 1. Poincaré recurrence vs. Birkhoff’s recurrence 3 1.1. Poincaré recurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincaré sequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: to...

Part 1. Analogies 3 1. Poincaré recurrence vs. Birkhoff’s recurrence 3 1.1. Poincaré recurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincaré sequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: to...