Mathematical Engineering, Politecnico di Milano. Milan.https://clainstone.comhttps://clainstone.com/threads/stochastic-dynamical-models/lecture-1-markov-chains-and-the-transition-matrixhttps://clainstone.com/threads/stochastic-dynamical-models/lecture-1-markov-chains-and-the-transition-matrixA Markov chain on a countable state space I is determined by an initial distribution λ and a stochastic matrix P. The probability of a path is the initial probability of its first state times a product of entries of P, the n-step transition matrix is the power P^n, and for a chain with two states P^n has a closed form.Sat, 19 Sep 2026 00:00:00 GMThttps://clainstone.com/threads/real-and-functional-analysis/lecture-1-cardinality-topology-and-measurable-spaceshttps://clainstone.com/threads/real-and-functional-analysis/lecture-1-cardinality-topology-and-measurable-spacesCardinalities are compared through injections, and \#mathcalP(X) > \#X for every set X. Topological and metric spaces are recalled, and every family of subsets generates a σ-algebra, the Borel σ-algebra when the family is a topology.Wed, 16 Sep 2026 00:00:00 GMT