It is well-known that the classification of flat surfaces in Euclidean 3space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane $C^2$ endowed with almost complex structure $J$, flat surfaces are the simplest ones from intrinsic point of views. On the other hand, from $J$-action point of views, the most natural surfaces in $C^2$ are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classifies flat slant surfaces in $C^2$. The main result states that, beside the totally geodesic ones, there are five large classes of flat slant surfaces in $C^2$. Conversely, every non-totally geodesic flat slant surfaces in $C^2$ is locally a surface given by these five classes.

We consider nonnegative solutions to the Cauchy problem for the quasilinear parabolic equations $u_t=\Delta u^m+K\left(x\right)u^p$ where $x\in R^N$, and $K\left(x\right)\ge 0$ has the following properties: $K\left(x\right)$- as $\left|x\right|\to \infty$ in some cone $D$ and $K\left(x\right)=0$ in the complement of $D$, where for $\sigma =-\infty$ we define that $K\left(x\right)$ has a compact support. We find a critical exponent $p_{m,\sigma }^{*}=p_{m,\sigma }^{*}\left(N\right)$ such that if $p\le p_{m,\sigma }^{*}$, then every nontrivial nonnegative solution is not global in time; whereas if $p>p_{m,\sigma }^{*}$ then there exits a global solution. We also find a second critical exponent, which is another critical exponent on the growth order $\alpha$ of the initial data $u_0\left(x\right)$ such that $u_0\left(x\right)$-$|x{|}^{-}\text{'}$ as $\left|x\right|\to \infty$ in some cone $D^{\prime }$ and $u_0\left(x\right)=0$ in the complement of $D^{\prime }$.

In this paper, we prove that depth zero representations are preserved by local theta correspondence for any type I reductive dual pairs over a $p$-adic field. Moreover, the minimal $K$-types of the paired depth zero irreducible admissible representations are paired by the theta correspondence for finite reductive dual pairs. As a consequence, we prove that the Iwahori-spherical representations are preserved by the local theta correspondence. Then we obtain some partial result of theta dichotomy for finite reductive dual pairs and $p$-adic reductive dual pairs of symplectic and orthogonal group, which is analogous to S. Kudla and S. Rallis' result for $p$-adic unitary groups.

Osserman pseudo-Riemannian manifolds with diagonalizable Jacobi operators are studied. A classification of such manifolds is achieved under two conditions on the number of different eigenvalues of the Jacobi operators and their associated elgenspaces.

The relationship between presentations of subshifts by Shannon graphs and by $\lambda$-graph systems is studied. A class of presentations of subshifts by 2-graph systems is characterized. A notion of synchronization is introduced. A class of presentations by $\lambda$-graph systems, that are specifically associated to subshifts that fall under this notion, is characterized.

We give a Fourier integral representation of harmonic functions in three variables in terms of the current of integration over $\{z_1^2+z_2^2+z_3^2=0\}\subset C^3$.

H. Miller introduced a filtration of Stiefel manifolds which splits stably and proved that the starata of the filtration are vector bundles, so that Stiefel manifolds are stably equivalent to a wedge of the corresponding Thom spaces. In this paper, we shall give a finer $p$-local stable splitting of the Stiefel manifolds by using the Adams operations.

By a recent method to estimate the derivatives $\left|w^{\left(k\right)}\right(z_i\left)\right|,$$k>1$, at certain $a$-points of a meromorphic function $w\left(z\right)$ in terms of the Ahlfors-Shimizu characteristic and of $\left|w^{\prime }\right(z_i\left)\right|$, we improve some classical results on the growth of meromorphic solutions of certain algebraic differential equations. Moreover, we offer similar results for equations involving inverse derivatives and derivatives of a power $w^t$ of a meromorphic function $w$.

Behavior of least-energy solutions to Matukuma type equations are discussed. Especially, how they vanish or blow up are investigated. In either case, the exact solution to a special equation plays a central role to analyze the behavior.

Making use of an integral operator $I^{\alpha }$ which was defined and studied earlier by Srivastava et al., the author introduces two novel families of strongly starlike functions $S{T}_{\alpha }(\beta ,\gamma )$ and $C{V}_{\alpha }(\beta ,\gamma )$. Certain properties of these classes are discussed.

In this paper we use an identity of Hardy-Stein type in investigations of the harmonic $H^p\left(B\right)$ and Bergman $b^p\left(B\right)$ spaces.

In this note, we corrected the coefficients of formula (2.12) in Proposition 1 and the third formula in Proposition 2 in our paper [1]. We should remark that our theorems in [1] are not harmed.