The aim of this paper is to prove the following theorem:

Theorem 1: The nilpotent injectors of $M(22)$ are Sylow $2$-subgroups.

Let $m, n$ be integers with $n\geq 2$, $m\geq 1$, the multilinear Riesz potential operators be defined by $$ I_{\alpha}^{(m)}({\bf{f}})(x) = \int_{(\mathbb{R}^{n})^{m}} \frac{f_1(y_1) \dots f_m(y_m)}{| (x-y_1, \dots, x-y_m) |^{mn-\alpha}}d{\bf{y}}, $$ where ${\bf{y}}=(y_1, \dots, y_m)$ and ${\bf{f}}=(f_{1}, \dots, f_{m})$. In the first part of this paper, the boundedness for the operator $I_{\alpha}^{(m)}$ on the homogeneous Herz-Morrey product spaces, $M\dot{K}_{p_1,q_1}^{n(1-1/q_1),\lambda_1}(\mathbb{R}^n) \times\dots\times M\dot{K}_{p_m,q_m}^{n(1-1/q_m),\lambda_m}(\mathbb{R}^n)$, and on the Herz-type Hardy product spaces, $H\dot{K}_{q_1}^{\sigma_1,p_1}(\mathbb{R}^n) \times\dots\times H\dot{K}_{q_m}^{\sigma_m,p_m}(\mathbb{R}^n)$ for $\sigma_i>n(1-1/q_i)$, are established respectively. The second goal of the paper is to extend the known $L^p$-bounded\-ness of $I_\alpha^{(m)}$ to generalized Morrey spaces, $L^{p,\phi}(\mathbb{R}^n)$, where $p\in[1,+\infty)$ and $\phi$ is the suitable doubling and integral functions.

We study a special type of invariant subspaces ${\mathcal M}$ on the bidisk which are studied in the previous papers. We determine the rank of cross commutators on $H^2\ominus {\mathcal M}$, and study when ${\mathcal M}$ is generated by ${\mathcal M}\ominus (z{\mathcal M}+w{\mathcal M})$ as an invariant subspace of $H^2$.

For a superbiharmonic function $u$ in the unit ball with the growth condition of spherical means, we show that $u$ is represented as the sum of a generalized Riesz potential and a biharmonic function. This representation is referred to as Riesz decomposition for superbiharmonic functions. \\ ~~~The superharmonic case is treated similarly.

We study differential geometry of spacelike surfaces in Anti de Sitter 3-space from the contact viewpoint. We define the timelike Anti de Sitter Gauss images and timelike Anti de Sitter height functions on spacelike surfaces and investigate the geometric meanings of singularities of these mappings. We consider the contact of spacelike surfaces with models (so-called AdS-great-hyperboloids) as an application of Legendrian singularity theory.

In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida [13] is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.

The construction of a tropical hypersurface is given by modeling the classical construction of a complex hypersurface. A tropical meromorphic function of finite type is shown to be a tropical rational function. One also has tropical nullstellensatz.

In this paper, we suppose that $M$ is a negatively curved $C_{K}-G$-manifold, $K\leq2$, and $M^{G}\neq\emptyset$. Then we characterize $M$ from topological view point.