Let X and Y be two complex manifolds, let$D$⊂$X$and$G$⊂$Y$be two nonempty open sets, let$A$(resp.$B$) be an open subset of ∂$D$(resp. ∂$G$), and let$W$be the 2-fold cross (($D$∪$A$)×$B$)∪($A$×($B$∪$G$)). Under a geometric condition on the boundary sets$A$and$B$, we show that every function locally bounded, separately continuous on$W$, continuous on$A$×$B$, and separately holomorphic on ($A$×$G$)∪($D$×$B$) “extends” to a function continuous on a “domain of holomorphy” $\widehat{W}$ and holomorphic on the interior of $\widehat{W}$ .

In this paper we consider the problem of the non-empty intersection of exposed faces in a Banach space. We find a sufficient condition to assure that the non-empty intersection of exposed faces is an exposed face. This condition involves the concept of$inner point$. Finally, we also prove that every minimal face of the unit ball must be an extreme point and show that this is not the case at all for minimal exposed faces since we prove that every Banach space with dimension greater than or equal to 2 can be equivalently renormed to have a non-singleton, minimal exposed face.

Estimation of longitudinal data covariance structure poses significant challenges because the data usually are collected at irregular time points. A viable semiparametric model for covariance matrixes has been proposed that allows one to estimate the variance function nonparametrically and to estimate the correlation function parametrically by aggregating information from irregular and sparse data points within each subject. But the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of parameters in the covariance model are largely unknown. We address this problem in the context of more general models for the conditional mean function, including parametric, nonparametric, or semiparametric. We also consider the possibility of rough mean regression function and introduce the difference-based method to reduce biases in the context of varying-coefficient partially linear mean regression

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of\KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality\KachruVafa\KachruSilverstein.

Spin polynomial invariants, as denned by Pidstrigatch and Tyurin, are used to define differentiable invariants for Dolgachev surfaces. In order to define and compute these invariants, some machinery is developed to study the wall structure for Spin polynomials, to deal with the fact that the moduli spaces have the wrong dimension and to handle nonreduced structures. The invariants are then computed.