The path complex and its homology were defined in the previous papers of authors. The theory of path complexes is a natural discrete generalization of the theory of simplicial complexes and the homology of path complexes provide homotopy invariant homology theory of digraphs and (nondirected) graphs. In the paper we study the homology theory of path complexes. In particular, we describe functorial properties of paths complexes, introduce notion of homotopy for path complexes and prove the homotopy invariance of path homology groups. We prove also several theorems that are similar to the results of classical homology theory of simplicial complexes. Then we apply obtained results for construction homology theories on various categories of hypergraphs. We describe basic properties of these homology theories and relations between them. As a particular case, these results give new homology theories on the category of simplicial complexes.

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We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases.

This paper introduces an efficient approach to integrating non-local statistics into the higher-order Markov Random Fields (MRFs) framework. Motivated by the observation that many non-local statistics (eg, shape priors, color distributions) can usually be represented by a small number of parameters, we reformulate the higher-order MRF model by introducing additional latent variables to represent the intrinsic dimensions of the higher-order cliques. The resulting new model, called NC-MRF, not only provides the flexibility in representing the configurations of higher-order cliques, but also automatically decomposes the energy function into less coupled terms, allowing us to design an efficient algorithmic framework for maximum a posteriori (MAP) inference. Based on this novel modeling/inference framework, we achieve state-of-the-art solutions to the challenging problems of class-specific image segmentation and template-based 3D facial expression tracking, which demonstrate the potential of our approach.