Lectures related to Vertical and horizontal bundles

Covers smooth maps, vector fields, and retractions on manifolds, emphasizing the importance of smoothly varying curves.

Covers continuous-time Markov chains and Kolmogorov equations in stochastic communication models.

Covers the equations of Kolmogorov for continuous-time Markov chains.

Covers the determination of matrices associated with linear transformations and explores the kernel and image concepts.