In this paper, we prove some properties and characterizations of stratifiable spaces and the following theorem:

Theorem. The following are equivalent:

1. $(Y,\tau)$ is a stratifiable space.

2. There is a zero-dimension submetric stratifiable space $(X, \mu)$ with $M_{3}$-structures and an irreducible perfect map $f:(X,\mu) \to (Y,\tau)$.

A stratifiable space $(X,\mu)$ is said to have an $M_3$-structure if $(X,\mu)$ satisfies the following conditions $A$ and $B$:

$A$: There is a countable collection $\mathscr{H} = \cup \mathscr{H}_{n}$ of $\rho$-closed sets such that:

1. $H(n',i') \subset H(n,i)$ or $\rho(H(n,i), H(n',i')) = r \gt 0$ if $H(n,i), H(n',i') \in \mathscr{H}$ with $n' \gt n$.

2. $\mathscr{H}_{n}$ is a partition of $X$ for each $n \in \mathbb{N}$.

$B$: There is a $g$-function $\mathscr{W}$ such that:

1. $\cap_{n}W(n,x) = \{x\}$.

2. $x \in W(n,x_{n})$, then $\{x_{n}: n \in \mathbb{N} \}$ converges to $x$.

3. If $H$ is closed and $x \notin H$, $x \notin Cl_{\mu}\left(\cup\{W(n,x'): x' \in H\}\right)$ for some $n$.

4. $x' \in W(n,x)$ implies $W(n,x') \subset W(n,x)$.

5. $H(n,i) \cap \left(\cup \mathscr{W}_{nj} \right) = \emptyset$ if $j \gt i$.

6. $W(n,x) \subset W(n-1,x)$.

7. Each $\mathscr{W}_{nm}$ is a $\rho$-discrete $\rho$-clopen collection.

8. $W(n,x) \subset c(n,x) \in \mathscr{C}$ for each $x \in X.$

Here $\mathscr{C}$ is a $g$-function of the stratifiable space $(X,\mu)$.

The role of countable paracompactness to obtain a (setvalued) selection avoiding extreme points is investigated. In particular, we prove the following: Let $X$ be a topological space, $Y$ a normed space and $\varphi$ a lower semicontinuous compact-and convexvalued mapping of $X$ to $Y$. If one of the following conditions is valid, then $\varphi$ admits a lower semicontinuous set-valued selection $\phi$ such that $\phi(x)$ is compact and convex, and each point of $\phi(x)$ is not an extreme point of $\varphi(x)$ for each $x \in X$; (1) the infimum of the set of all diameters of $\varphi(x)$ with $x \in X$ is positive, (2) X is countably paracompact and the cardinality of $\varphi(x)$ is more than one for each $x \in X$. We also give characterizations of some topological spaces in terms of (set-valued) selections avoiding extreme points.

In this paper, we investigate the relationship between differential polynomials and meromorphic functions of finite order of some second order linear differential equations with meromorphic coeffcients. We obtain some precise estimates.

In this paper, we show that a graph $G$ contains no 4-cycles if and only if $\Vert G \Vert$ is a strong $\mathbb{Z}_2$-deformation retract of the box complex $\Vert B(G) \Vert$ of $G$, where $G$ is the 1-dimensional free simplicial $\mathbb{Z}_{2}$-complex introduced in [2].

A complete classification for a compact Kaehler submanifold $M_n$ in $P_{n+p}(\mathbb{C})$ with the scalar curvature $\rho \geq n^{2}$ is given, so that a conjecture of K. Ogiue is resolved partially.

For one dimensional tilings, we can define associated groups. And it is known that those groups has Gauss decomposition. We will show one characterization of those groups.

We study the lightlike geometry of the second fundamental form of the intersection between an algebraic hypersurface in $\mathbb{C}_{s}^{n}$ and a pseudosphere $S_{2s}^{2n-1}(2/\sqrt{k})$ including a class of extrinsic spheres which are not homotopy spheres.

Subject to the Riemann hypothesis for Dirichlet $L$ functions an asymptotic formula is obtained for the number of representations of a natural number $n$ as the sum of a prime and a $k$-th power, valid for almost $n$. Estimates for the error term in the asymptotic formula as well as for the size of the exceptional set are of a smaller order of magnitude than was known previously.

In [1], J. Berndt and H. Tamaru classified all the cohomogeneity one actions on Riemannian symmetric spaces of noncompact type with a totally geodesic singular orbit. Also they provided that there is a one-to-one correspondence between the totally geodesic singular orbits of cohomogeneity one actions on a Riemannian symmetric space of noncompact type and those on its dual simply connected compact Riemannian symmetric space. In this paper, we determine stability of the totally geodesic singular orbits in simply connected compact symmetric spaces which obtained by the duality stated above.

Let $R_{s,k}(n)$ denote the number of representations of $n$ as the sum of $s$ positive integral $k$th powers. We develop methods to establish asymptotic formulae for moments of $R_{s,k}(n)$ in which $n$ is restricted to a thin sequence consisting of values of a given polynomial. As a first example, we discuss the case of cubes, and present conclusions for sums of 6 and 7 cubes in which the polynomial is quadratic, and for sums of 7 cubes in which the polynomial is cubic. We also consider the case wherein the exponent $k$ is large, and briefly describe corresponding results for the binary Goldbach problem.