Neighbor discovery is a fundamental problem in wireless networks. In this paper, we study asynchronous neighbor discovery between duty-cycled mobile devices. Each node is duty-cycled, i.e., its radio may only be active for a small fraction of the time. The duty cycles of the nodes can be the same or different, leading to symmetric or asymmetric cases of the neighbor discovery problem. In addition, the setting is asynchronous, i.e., clocks of different nodes may not be synchronized. Most existing studies assume an integer model (where time proceeds in discrete steps); two recent studies break away from this assumption, which allows them to develop significantly more efficient schemes. Our study improves the state-of-the-art in three main fronts. Firstly, we develop a generalized non-integer model (where time is continuous) that permits unified treatment of the assumptions in existing studies. We also provide a reduction that transforms any schedule in the basic integer model to a c...