We show that a compact operator A is a multiple of a positive semi-definite operator if and only if (A B) W (A) W (B), for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold.

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.

Askewing'method is shown to effectively reduce the order of bias of locally parametric estimators, and at the same time retain positivity properties. The technique involves first calculating the usual locally parametric approximation in the neighbourhood of a point x'that is a short distance from the place x where the we wish to estimate the density, and then evaluating this approximation at x. By way of comparison, the usual locally parametric approach takes x'= x. In our construction, x'-x depends in a very simple way on the bandwidth and the kernel, and not at all on the unknown density. Using skewing in this simple form reduces the order of bias from the square to the cube of bandwidth; and taking the average of two estimators computed in this way further reduces bias, to the fourth power of bandwidth. On the other hand, variance increases only by at most a moderate constant factor.

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