In this Note we prove that, under a weaker condition than the-lemma, the existence of balanced metrics is preserved under small deformations. This weaker condition is satisfied on the twistor space over a compact self-dual four manifold.

We introduce a technique based on Gaussian maps to study whether a surface can lie on a threefold as a very ample divisor with given normal bundle. We give applications, among which one to surfaces of general type and another to Enriques surfaces. In particular, we prove the genus bound g  17 for Enriques Fano threefolds. Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admit Q-smoothings.

Finding correspondences between two surfaces is a fundamental operation in various applications in computer graphics and related fields. Candidate correspondences can be found by matching local signatures, but as they only consider local geometry, many are globally inconsistent. We provide a novel algorithm to prune a set of candidate correspondences to those most likely to be globally consistent. Our approach can handle articulated surfaces, and ones related by a deformation which is globally non-isometric, provided that the deformation is locally approximately isometric. Our approach uses an efficient diffusion framework, and only requires geodesic distance calculations in small neighbourhoods, unlike many existing techniques which require computation of global geodesic distances. We demonstrate that, for typical examples, our approach provides significant improvements in accuracy, yet also reduces time and memory costs by a factor of several hundred compared to existing pruning techniques. Our method is furthermore insensitive to holes, unlike many other methods.

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We study the estimation of the additive components in additive regression models, based on the weighted sample average of regression surface, for stationary <i></i>-mixing processes. Explicit expression of this method makes possible a fast computation and allows an asymptotic analysis. The estimation procedure is especially useful for additive modeling. In this paper, it is shown that the average surface estimator shares the same optimality as the ideal estimator and has the same ability to estimate the additive component as the ideal case where other components are known. Formulas for the asymptotic bias and normality of the estimator are established. A small simulation study is carried out to illustrate the performance of the estimation and a real example is also used to demonstrate our methodology.