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In this paper, we study the Atiyah class and Todd class of the DG manifold $(F[1],d_F)$ corresponding to an integrable distribution $F \subset T_\mathbb{K} M = TM \otimes_{\mathbb{R}} \mathbb{K}$, where $\mathbb{K} = \mathbb{R}$ or $\\mathbb{C}$. We show that these two classes are canonically identical to those of the Lie pair $(T_{\mathbb{K}} M, F)$. As a consequence, the Atiyah class of a complex manifold $X$ is isomorphic to the Atiyah class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$. Moreover, if $X$ is a compact K\"{a}hler manifold, then the Todd class of $X$ is also isomorphic to the Todd class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$.

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.

The Hamiltonian cycle problem is an important problem in graph theory and computer science. It is a special case of the famous traveling salesman problem. A signicant amount of research has been done on the special cases of nding Hamiltonian cycles in subgraphs of triangular, square and hexagonal grids. However, there is little work on more complicated grids. In this paper, we investigate the hardness of Hamiltonian cycle problem in grid graphs of semiregular tessellations, which are tessellations formed by two or more kinds of regular polygons. There are only eight semiregular tessellations, and we prove that the Hamiltonian cycle problem in all of them are NP-complete by reducing from NP-complete problems, such as the Hamiltonian cycle problem in max degree 3 bipartite planar graphs. Knowing the NP-completeness of Hamiltonian cycle problem in semiregular grids indicates that there will not be any polynomial time algorithm that solves the Hamiltonian path problem in these tessellations if NP does not equal to P, which helps show the limits of ecient motion planning algorithms and provides new information about what makes problems computationally dicult to solve.

The goal of this paper is to develop a novel semi-implicit spectral deferred correction (SDC) time marching method. The method can be used in a large class of problems, especially for highly nonlinear ordinary differential equations (ODEs) without easily separating of stiff and non-stiff components, which is more general and efficient comparing with traditional semi-implicit SDC methods. The proposed semi-implicit SDC method is based on low order time integration methods and corrected iteratively. The order of accuracy is increased for each additional iteration. And we also explore its local truncation error analytically. This SDC method is intended to be combined with the method of lines, which provides a flexible framework to develop high order semi-implicit time marching methods for nonlinear partial differential equations (PDEs). In this paper we mainly focus on the applications of the nonlinear PDEs with higher order spatial derivatives, e.g. convection diffusion equation, the surface diffusion and Willmore flow of graphs, the Cahn– Hilliard equation, the Cahn–Hilliard–Brinkman system and the phase field crystal equation. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the fully discrete schemes are all high order accurate in both space and time, and stable numerically with the time step proportional to the spatial mesh size. Numerical experiments are carried out to illustrate the accuracy and capability of the proposed semi-implicit SDC method.