Let $F: \Sigma \to \mathbb{R}^3$ be a Blaschke immersion of an affine surface $(\Sigma,\nabla)$ with a positive definite affine fundamental form such that $dim Im \,\ R = 1$ where $R$ is the curvature tensor. Suppose that there exists another immersion of the same surface with the same induced affine connection $\nabla$ which is not affine equivalent to the first one. Then we give explicitely $F$. Therefore all immersions which admit another immersion which is not affine equivalent to the original one are classified.

Let $\{n_k\}_{k=1}^\infty$ be a sequence of positive integers with Hadamard gap. For an analytic function $F(z) = \sum_{n=0}^{\infty}a_n z^n$ in the unit disc satisfying $\sup_{0 \lt r \lt 1}$ $\int_0^{2\pi} |F(re^{i\theta})|\, d\theta \lt \infty$, the inequality $( \sum_{k=1}^{\infty}|a_{n_k}|^2 )^{1/2} \lt \infty$ holds, which is familiar as Paley's inequality. In this paper, an integral transform version of this inequality is established.

Let $G$ be a finite abelian subgroup of $PGL(r1,K)=\mathrm{Aut}(\mathbb{P}^{r-1}_K)$. In this paper, we prove that the normalization of the $G$-Hilbert scheme$Hilb^G(\mathbb{P}^{r-1})$ is described as a toric variety, which corresponds to the Gröbner fan for some homogeneous ideal $I$ of $K[x_1, \ldots ,x_r]$.

We will prove that if a finite abelian group is a direct product of its subsets such that one subset has four elements and the others have prime cardinalities, then at least one of the factors must be periodic.

In this paper, we show the boundedness of pseudo-differential operators of class $S_{0,0}^m$ on the Herz spaces $\dot{K}_q^{\alpha,p}$ and the Herz-type Hardy spaces $H\dot{K}_q^{\alpha,p}$.

The paper studies the counting functions of the simple and multiple zero points of meromorphic functions that share three values with finite weight. The results in this paper improve some results of H.X. Yi, X.M. Li and H.X. Yi and other authors. Examples show that the results in this paper are best possible.

Our aim in this article is to study the long time behavior, in terms of finite-dimensional attractors, of doubly nonlinear Cahn-Hilliard type equations. In particular, we prove the existence of an exponential attractor. We also study the continuity of exponential attractors when the equations converge to the usual Cahn-Hilliard equation.

A surface in the Lorentzian complex plane ${\bf C}^2_1$ is called {\it marginally trapped\/} if its mean curvature vector is light-like at each point on the surface. In this article, we classify marginally trapped surfaces of constant curvature in the Lorentzian complex plane ${\bf C}^2_1$. Our main results state that there exist twenty-one families of marginally trapped surfaces of constant curvature in ${\bf C}^2_1$. Conversely, up to rigid motions and dilations, marginally trapped surfaces of constant curvature in ${\bf C}^2_1$ are locally obtained from these twenty-one families.