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In this article we provide a proof of the Iitaka Cnm conjecture for algebraic fiber spaces over tori.

A bandwidth selection method is proposed for local linear regression. Our approach is to combine the ideas of optimal bandwidth selection of Hall et al.(1991) in kernel density estimation, and use of direct bias and variance in Fan and Gijbels (1995) for local linear regression. We show that the bandwidth selector has an optimal relative rate of convergence of n-1/2 with n the sample size.

The nth power of the volume functional Vnof polytopes P in R^n, according to dimensions of the spaces spanned by any nunit outer normal vectors of P, is decomposed into nhomogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R^3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

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