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Large language models(LLMs) have shown its outperforming ability on various tasks and question answering. However, LLMs require substantial memory storage on low-resource devices. More critically, the computational speed on these devices is also severely limited. In this paper, we propose SPA(Side Plugin Adaption), a lightweight architecture for fast on-devices inference on the constraints of strict on-devices computation and memory constraints. Compared with other on-devices seq2seq generation, SPA could make a fast and stable inference on low-resource constraints, allowing it to obtain cost effiency. Our method establish an interaction between a pretrained LLMs on-cloud and additive parameters on-devices, which could provide the knowledge on both pretrained LLMs and featured personal feature. Further more, SPA provides a framework to keep feature-base parameters on low computational devices while leave the parameters containing general information on the high computational devices., Comment: 12 pages, third version of SPA(Side Plugin Adaption)

In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough., Comment: 15 pages. Some typos are corrected. Comments are welcome