We consider the Schrödinger equation for the harmonic oscillator$i$$∂$_{$t$}$u$=$Hu$, where$H$=−Δ+|$x$|^{2}, with initial data in the Hermite–Sobolev space$H$^{−$s$/2}$L$^{2}(ℝ^{$n$}). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems.

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

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We develop a method searching for quasi-conformal maps from point-cloud surfaces to Euclidean plane. The maps can be got by directly solving Beltrami equation with proper boundary condition or solving optimization problem of the variational formu- lation. The point integral methd (PIM) is used for discretization on point clouds. Numerical experiments suggest that the method converges as the density of points in- creases.