Let the Paneitz operator $P_{0}$ be strictly positive on a closed 3-manifold $M$ with a fixed conformal class. It is proved that the solution of a fourth-order $Q$-curvature flow exists on $M$ for all time and converges smoothly to a metric of constant $Q$-curvature.

This paper treats a holomorphic self-mapping $f: \Omega \rightarrow \Omega$ of a bounded domain $\Omega$ in a separable Hilbert space ${\cal H}$ with a fixed point $p$. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Carath\'eodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping $f$ and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.

Motivated by a recent work of Loaiza {\it et al.} for the holomorphic case on the disk, we introduce and study the notion of Schatten-Herz type Toeplitz operators acting on the harmonic Bergman space of the ball. We obtain characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions. Our characterization in terms of Berezin transforms settles a question posed by Loaiza {\it et al.}

We prove that the k-truncated microsupport of the specialization of a complex of sheaves $F$ along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of $F$. In the complex case, applying our estimates to $F=R\cal{H}\text{om}_{\cal{D}}(\cal{M}, \sho)$, where $\cal{M}$ is a coherent $\cal{D}$-module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When $\cal{M}$ is regular along $Y$ we also obtain estimates for the truncated microsupport of the holomorphic solutions of the induced system along $Y$ as well as for the nearby-cycle sheaf of $\shm$ when $Y$ is a hypersurface.

Let $k$ a regular noetherian $\mathbb{F}_p$-algebra, let $A = k[x,y]/(xy)$ be the coordinate ring of the coordinate axes in the affine $k$-plane, and let $I = (x,y)$ be the ideal that defines the intersection point. We evaluate the relative $K$-groups $K_q(A,I)$ completely in terms of the big de Rham-Witt groups of $k$. This generalizes a formula for $K_1(A,I)$ and $K_2(A,I)$ by Dennis and Krusemeyer.

For a finite set $S = \{a_1, \ldots, a_q\}$, consider the polynomial $P_S(w) = (w - a_1)(w - a_2) \cdots (w - a_q)$ and assume that $P'_S(w)$ has distinct $k$ zeros. Suppose that $P_S(w)$ is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functions $\phi$ and $\psi$, the equality $P_S(\phi) = c P_S(\psi)$ implies $\phi = \psi$, where $c$ is a nonzero constant which possibly depends on $\phi$ and $\psi$. Then, under the condition $q > k + 2$, we prove that, for any given nonconstant entire function $g$, there exist at most $(2q\!-\!2)/(q-k-2)$ \linebreak nonconstant entire functions $f$ with $f^*(S) = g^*(S)$, where $f^*(S)$ denotes the pull-back of $S$ considered as a divisor. Moreover, we give some sufficient conditions of uniqueness polynomials for entire functions.

We show that the Hilbert-Kunz multiplicities of the reductions to positive characteristics of an irreducible projective curve in characteristic 0 have a well-defined limit as the characteristic tends to infinity.

For a bounded domain $\Omega$ on the plane we show the inequality $c_{\Omega}(z)^{2} \leq 2\pi K_{\Omega}(z)$, $z \in \Omega$, where $c_{\Omega}(z)$ is the logarithmic capacity of the complement $\mathbb{C} \setminus \Omega$ with respect to $z$ and $K_{\Omega}$ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality $c_{\Omega}(z)^{2} \leq \pi K_{\Omega}(z)$ conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.

For all $n > 1$, the characteristic function of the unit ball in $\mathbb{R}^{2n}$ is not the symbol of a bounded bilinear multiplier operator from $L^{p}(\mathbb{R}^{n}) \times L^{q}(\mathbb{R}^{n})$ to $L^{r}(\mathbb{R}^{n})$ when $1/p+1/q = 1/r$ and exactly one of $p$, $q$, or $r' = r/(r-1)$ is less than $2$.

Applying the $L^2$ method of solving the $\bar{\partial}$-equation, it is shown that compact K\"{a}hler manifolds of dimension $\geq 3$ admit no Levi flat real analytic hypersurfaces whose complements are Stein.

In the present paper, it is proved that the $K_{0}$-group of a Toeplitz algebra on any strongly pseudoconvex domain is always isomorphic to the $K_{0}$-group of the relative continuous function algebra, and is thus isomorphic to the topological $K^{0}$-group of the boundary of the relative domain. Further there exists a ring isomorphism between the $K_{0}$-groups of Toeplitz algebras and the Chern classes of the relative boundaries of strongly pseudoconvex domains. As applications of our main result, $K$-groups of Toeplitz algebras on some special strongly pseudoconvex domains are computed. Our results show that the Toeplitz algebras on strongly pseudoconvex domains have rich structures, which deeply depend on the topological structures of relative domains. In addition, the first cohomology groups of Toeplitz algebras are also computed.