We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi [H]) via vacant double groupoids as explained in [AN]. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.

The homogenized ideal dual complex of an arbitrary meet-semilattice is introduced and described explicitly. Meet-distributive meet-semilattices whose homogenized ideal dual complex is level are characterized.

In this paper, the authors get the $L^{p}$ estimates for the commutators generated by strongly singular integral operators and ${\rm BMO}$ functions and the corresponding multilinear operators by the scale changing method introduced by Carleson and Sjölin.

In this paper, based on a Harnack-type estimate and a local Sobolev constant bounded for the Calabi flow on closed surfaces, we extend author's previous results and show the long-time existence and convergence of solutions of 2-dimensional Calabi flow on closed surfaces. Then we establish the uniformization theorem for closed surfaces.

In this paper, using techniques of value distribution theory, we give a uniqueness theorem for meromorphic mappings of $\mathbb{C}^{m}$ into $\mathbb{C}P^{n}$ with $(3n+1)$ moving targets and truncated multiplicities.

Let $g(f)$, $S(f)$, $g_{\lambda}^{*}(f)$ be the Littlewood-Paley $g$ function, Lusin area function, and Littlewood-Paley $g_{\lambda}^{*}$ function of $f$, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a $\mathrm{BMO}$ function $f$, one of the above functions is finite for a single point in $\mathbb{R}^{n}$, then it is finite a.e. on $\mathbb{R}^{n}$, and $\mathrm{BMO}$ boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, $\mathrm{BMO}$, and Lipschitz spaces). One of us improved his $g_{\lambda}^{*}$ result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel $\mu^{\rho}(f)$, $\mu_{S}^{\rho}(f)$ and $\mu_{\lambda}^{\ast, \rho}(f)$. In this paper, we show that the same results hold also in the case of rough kernel satisfying $L^{p}$-Dini type condition.

In this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds with $S^{1}$-action.