Abstract. In this paper, we study stochastic differential equations with two well-separated time scales. We prove that the rate of strong convergence to the averaged effective dynamics is of order O(ε 1/2 ) , where ε ≪ 1 is the parameter measuring the disparity of the time scales in the system. The convergence rate is shown to be optimal through examples.

We report a biomimetic mineralization of hydroxyapatite induced by poly-dopamine-functionalized reduced graphene oxide (RGO-PDA). Graphene oxide was first simultaneously reduced and surface functionalized by one-step oxidative polymerization of dopamine. The resultant RGO-PDA was further used as a bioinspired surface to mimic the mineralization of hydroxyapatite during bone formation. MC3T3-E1 cells were cultured on the RGO-PDA substrates to observe various cellular activities and hydroxyapatite mineralization. The MC3T3-E1 cells on RGO-PDA substrates show higher cellular activities such as proliferation, adhesion, and osteogenic differentiation over the bare glass and graphene oxide substrates. Those results suggest the potentials of using RGO-PDA as osteogenesis-promoting scaffolds for successful applications in bone tissue regeneration.

In this paper, a spectral method is formulated as a numerical solution for the stochastic Ginzburg-Landau equation driven by space-time white noise. The rates of pathwise convergence and convergence in expectation in Sobolev spaces are given based on the convergence rates of the spectral approximation for the stochastic convolution. The analysis can be generalized to other spectral methods for stochastic PDEs driven by additive noises, provided the regularity condition for the noises.

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