The sheet number of a $2$-knot is a quantity which reflects the complexity of the knotting in $4$-space. The aim of this note is to determine the sheet numbers of the $2$- and $3$-twist-spun trefoils. For this purpose, we give a lower bound of the sheet number by the quandle cocycle invariant of a $2$-knot, and an upper bound by the crossing number of a $1$-knot.

Let $p \in (0, \infty)$ and ${\mathfrak B}_{1}^{p} $ be the class of analytic functions $f$ in the unit disk ${\mathbb D}$ with $f(0)=0$ satisfying $|f'(z)| \leq 1/(1-|z|^2)^p$. For $z_0, z_1 \in {\mathbb D}$, $w_1 \in {\mathbb C}$ with $z_0 \not= z_1$ and $|w_1| \leq 1/(1-|z_1|^2)^p$, put $V^p(z_0 ; z_1,w_1) $ be the variability region of $f'(z_0)$ when $f$ ranges over the class ${\mathfrak B}_1^p$ with $f'(z_1) = w_1$, i.e., $V^p(z_0 ; z_1,w_1) = \{ f'(z_0) : f \in {\mathfrak B}_1^p \; \textrm{and} \; f'(z_1) = w_1 \}$. In 1988 M. Bonk showed that $V^1(z_0 ; z_1,w_1) $ is a convex closed Jordan domain and determined it by giving a parametrization of the simple closed curve $\partial V^1(z_0 ; z_1,w_1) $. He also derived distortion theorems for ${\mathfrak B}_{1}^1$ as corollaries. In the present article we shall refine Bonk's method and explicitly determine $V^p(z_0 ; z_1,w_1)$.

Hopf surfaces with the infinite cyclic fundamental group are said to be primary and their moduli spaces are studied by K. Dabrowski. We extend his results to the moduli spaces for the secondary Hopf surfaces with abelian fundamental groups which are diffeomorphic to $L(p,q) \times S^1$. The punctured disk $D^*$ is the moduli space for the other cases.

In this paper, we establish a fundamental asymptotic result for the solutions to second order linear nonautonomous neutral delay differential equations. By the use of a solution of the corresponding generalized characteristic equation, we transform the second order neutral delay differential equation into a first order neutral delay differential equation and then we utilize its generalized characteristic equation.

Let $\mu$ be a Radon measure on $\rd$ which satisfies $\mu(B(x,\,r))\leq Cr^{n}$ for any $x\in \rd$ and $r>0$ and some fixed positive constants $C$ and $n$ with $0<n\leq d$. In this paper, the relationship between the $L^p(\mu)$ boundedness and certain weak type endpoint estimates are considered for the singular integral operators.

In this paper, we compute the fundamental group of the complement of linear torus curves of maximal contact and we show that it is isomorphic to that of generic linear torus curves. As an application, we give new two Zariski triples.

For two meromorphic functions sharing two one-point sets and two three-point sets CM, we consider when one of them is a Möbius transform of the other.

Let $\boldsymbol{Y}=\left(Y_1, Y_2,\dots, Y_k\right)'$ be a random vector with multinomial distribution. In this paper we investigate the convergence rate of so-called power divergence family of statistics $\{I^\lambda(\boldsymbol{Y}),\lambda\in\mathbb{R}\}$ introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every $k\ge4$

$\Pr(2nI^\lambda(\boldsymbol{Y})<c)=G_{k-1}(c)+O(n^{-1+\mu(k-1)}),$

where $G_r(c)$ is the distribution function of chi-square random variable with $r$ degrees of freedom, $\mu(r)={6}/{(7r + 4)}$ for $3\le r\le 7$, $\mu(r)={5}/{(6r+2)}$ for $r\ge 8$. This refines Zubov and Ulyanov's result (2008). The proof uses Krätzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.