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We consider the multiple-access relay channel with two source nodes, one relay node and one destination node. For practical simplicity, we consider orthogonal transmission of the source messages and half-duplex relaying. Further, we assume that the relay is memoryless and is implemented based on a two-to-one deterministic mapping. Essentially the proposed mappings combine the two incoming analog signals and forward them to the destination, thus we term them analog network coding mappings. We investigate both linear and non-linear mappings. In particular, we propose to use mappings based on the Archimedean spiral for analog non-linear combining. In addition, we also propose to couple spiral mappings with sawtooth-like mappings to exploit the potential side information provided by the direct links. We investigate the resulting achievable rate regions and sum rates, and demonstrate significant gains.