Discrete Differential Calculus on Simplicial Complexes and Constrained Homology

Let $V$ be a finite set. Let $\mathcal{K}$ be a simplicial complex with its vertices in $V$. In this paper, we discuss some differential calculus on $V$. We construct some constrained homology groups of $\mathcal{K}$ by using the differential calculus on $V$. Moreover, we define an independent hypergraph to be the complement of a simplicial complex in the complete hypergraph on $V$. Let $\mathcal{L}$ be an independent hypergraph with its vertices in $V$. We construct some constrained cohomology groups of $\mathcal{L}$ by using the differential calculus on $V$.