This paper presents a study on graphene platelet (GPL)-reinforced alumina (Al2O3) ceramic composites and the relationships between the loading of GPL and both mechanical properties and in vitro biocompatibility. Al2O3 powders with different GPL contents were prepared and sintered using a gas protected pressure-less furnace. The examination of the results shows the density of the composites varying from 99.2% to 95.6% with the loading of GPL from 0.75 to 1.48 vol %. Raman studies show that moderate agglomerations of GPLs occur during the ball milling process and graphitic defects were produced during the high temperature processing. Mechanical properties of the Al2O3 matrix are significantly improved by adding GPLs. A maximum increase of approximately 60% in flexural strength and 70% in fracture toughness are achieved by introducing 0.75 vol % GPLs. In the biocompatibility tests, it was found that cells directly seeding on top of GPL/Al2O3 samples showed better initial attachment (3 h after seeding) and viability (3 days after incubation) than the monolithic Al2O3, indicating that the GPL/Al2O3 composites have comparable or more favorable biocompatibility. The excellent mechanical and biomedical properties of the GPL/Al2O3 composites may enable them to be applied to a wide range of engineering and biomedical applications.

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Suppose a matrix A ∈ R m×n of rank k with singular value decomposition A = U A Σ A V T A , where U A ∈ R m×k , V A ∈ R n×k are orthonormal and Σ A ∈ R k×k is a diagonal matrix. The statistical leverage scores of a matrix A are the squared row-norms defined by ℓ i = (U A ) i,: 2 2 , where i ∈ [m], and the matrix coherence is the largest statistical leverage score. These quantities play an important role in machine learning algorithms such as matrix completion and Nyström-based low rank matrix approximation as well as large-scale statistical data analysis applications. The best known classical algorithm to approximate these values runs in time O((mn + n 3 )log m) in [P. Drineas, M. Magdon-Ismail, M. W. Mahoney and D. P. Woodruff. Fast approximation of matrix coherence and statistical leverage. J. Mach. Learn. Res., (2012)13: 3475-3506]. In this work, inspired by recent development on dequantization techniques, we propose a fast novel classical algorithm for approximating the statistical leverage scores. Our novel algorithm has query and time complexity O poly k, κ, 1 ǫ , 1 δ , log(mn) , where κ is the condition number of A, and δ is the failure probability.

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