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Two metal-free porphyrins H[Formula: see text]Pp (a, b) and four metalloporphyrins CuPp (a, b) and ZnPp (a, b) have been applied to lithium/thionyl chloride (Li/SOCl[Formula: see text] battery as the high efficient catalysts for the first time. Notably, the battery discharge time in blank is only 1080 s while that of Li/SOCl2 battery after adding porphyrin/metalloporphyrins is 1980 s for H[Formula: see text]Pp (a), 2160 s for H[Formula: see text]Pp (b), 1560 s for CuPp (a), 1320 s for CuPp (b), 1620 s for ZnPp (a), and 1680 s for ZnPp (b), respectively. Furthermore, the energy of the battery whose electrolyte contains the H[Formula: see text]Pp (a, b), CuPp (a, b) and ZnPp (a, b) increases by approximately 11.88–77.14% than that of the battery without them. And the metal-free porphyrins H[Formula: see text]Pp (a) or H[Formula: see text]Pp (b) exhibit higher catalytic activities than that of metalloporphyrins CuPp (a, b) and ZnPp (a, b). In addition, the possible reasons to explain the excellent results were also analyzed.

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We address the load balance, referred as "fairness" in our work, in the relay selection procedure of the dual hop decode-and-forward (DF) cooperative networks. Firstly, the unfairness caused by the known bestpath relay selection (BRS) scheme is analyzed by the metric of blocking probability at the relays with equal relay channels. Secondly, aiming to improve the fairness while keeping a satisfied and comparable outage performance, we propose a fair relay selection algorithm based on mapping functions, including the signal to noise ratio (SNR) mapping functions and load mapping functions. It is worth mentioning that BRS scheme can be viewed as a special case of the proposed algorithm. Afterwards, the outage probability of the proposed algorithm is derived. During the derivation, the mixture Gamma distribution is used to model SNRs of wireless fading channels, so this kind of derivation is valid for different fading channels, e.g. Rayleigh, Nakagami-m etc.. Finally, taking the average load and outage probability as performance metrics, we simulated the proposed algorithm and compared it with the proportional fair cooperation (PFC) algorithm in the literature. As observed in the simulation results, the proposed algorithm guarantees fairness well among the relays, especially when the network is heavily burdened.

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