Abstract: This lecture is an appendix to the previous presentations about Hodge decomposition. 1. I will review the Helmholtz -Hodge decomposition and Laplacians for 2D polyhedral surfaces after Max Wardetzky and discuss the case of 3D polyhedral manifolds . 2. I will discuss how to use the linear algebra of Hodge decomposition to provide optimal size cycle realization of homology classes for a simplicial complexes with positive weights on simplexes. 3.I intend to formulate a number of simple but in my view interesting computational problems of interest in data analysis.