A Unified Relay Framework with both D-F and C-F Relay Nodes

Decode-and-forward (D-F) and compress-and-forward (C-F) are two fundamentally different relay strategies proposed by (Cover and El Gamal, 1979). Individually, either of them has been successfully generalized to multi-relay channels. In this paper, to allow each relay node the freedom of choosing either of the two strategies, we propose a unified framework, where both the D-F and C-F strategies can be employed simultaneously in the network. It turns out that, to fully incorporate the advantages of both the best known D-F and C-F strategies into a unified framework, the major challenge arises as follows: For the D-F relay nodes to fully utilize the help of the C-F relay nodes, decoding at the D-F relay nodes should not be conducted until all the blocks have been finished; However, in the multi-level D-F strategy, the upstream nodes have to decode prior to the downstream nodes in order to help, which makes simultaneous decoding at all the D-F relay nodes after all the blocks have been finished inapplicable. To tackle this problem, nested blocks combined with backward decoding are used in our framework, so that the D-F relay nodes at different levels can perform backward decoding at different frequencies. As such, the upstream D-F relay nodes can decode before the downstream D-F relay nodes, and the use of backward decoding at each D-F relay node ensures the full exploitation of the help of both the other D-F relay nodes and the C-F relay nodes. The achievable rates under our unified relay framework are found to combine both the best known D-F and C-F achievable rates and include them as special cases

Entire curves avoiding given sets in C^n

Let $F\subset\Bbb C^n$ be a proper closed subset of $\Bbb C^n$ and $A\subset\Bbb C^n\setminus F$ at most countable ($n\geq 2$). We give conditions of $F$ and $A$, under which there exists a holomorphic immersion (or a proper holomorphic embedding) $\phi:\Bbb C\to\Bbb C^n$ with $A\subset\phi(\Bbb C)\subset\Bbb C^n\setminus F$.Comment: 10 page