We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.Comment: Published in at http://dx.doi.org/10.1214/10-AOS799 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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We prove a small excess regularity theorem for almost minimizers of a quasiconvex variational integral of subquadratic growth. The proof is direct, and it yields an optimal modulus of continuity for the derivative of the almost minimizer. The result is new for general almost minimizers, and in the case of absolute minimizers it considerably simplifies the existing proof. Mathematics Subject Classification (2000). 49N60, 26B25

Selective Laser Melting (SLM) was used to fabricate scaffolds using the titanium alloy Ti-6Al-4V. Two types of high porosity open-cell structures were manufactured: the first built from topology optimised designs with maximised stiffness, and the second from gyroid labyrinths. In mechanical compression tests the scaffolds demonstrate exceptional strength- and stiffness-to-weight ratios. In particular, for densities in the range 0.2-0.8g/cm3 the topology optimised scaffolds have specific strength and stiffness that are superior to those of comparable materials in the literature. In addition, the optimised scaffolds have the benefit of being elastically isotropic. The results of finite element calculations accurately match the measured stiffness of the scaffolds. Calculated strain energy distributions provide insight into how the high stiffness and strength of the optimised designs is connected to their efficient distribution of load