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This paper introduces gradient, adjoint, and $p$-Laplacian definitions for oriented hypergraphs as well as differential and averaging operators for unoriented hypergraphs. These definitions are used to define gradient flows in the form of diffusion equations with applications in modelling group dynamics and information flow in social networks as well as performing local and non-local image processing., Comment: 33 pages, 5 figures, submitted to Scale Space and Variational Methods, part of it published in International Conference on Scale Space and Variational Methods in Computer Vision proceedings