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We introduce the notions of a path complex and its homologies. Particular cases of path homologies are simplicial homologies and digraph homologies. We state and prove some properties of path homologies, in particular, the K¨unneth formulas for Cartesian product and join, which happen to be true at the level of chain complexes.

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We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvaturelike quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.

Given a compact Riemannian manifold $M$, we consider a warped product manifold $\bar M = I \times_h M$, where $I$ is an open interval in $\Bbb R$. For a positive function $\psi$ defined on $\bar M$, we generalize the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $\sigma_k(\kappa)=\psi(V,\nu(V))$. We also obtain some existence results for the starshaped compact hypersurface $\Sigma$ satisfying the above equation with various assumptions.