We will give a homomorphism of the cohomology rings $H^{*}\left(\Gamma ,\Gamma \right)\to H^{*}\left(Q_{8,\psi }\Gamma \right)$induced by the ring homomorphism from the integral group algebra $\Lambda =\mathbb{Z}{Q}_8$ of the quaternion group $Q_8$ to the quaternion algebra $\Gamma =\Lambda e$ for a central idempotent $e$ in $\mathbb{Q}{Q}_8$.

The number of the solitons of the Korteweg-de Vries (KdV) equation is considered when the initial value of the solution is given. Upper and lower bounds for the number of solitons of this equation are obtained under some conditions on the initial value of the solution.

Let $K$ be a function field over a field $\kappa$ of characteristic and let $R$ be a discrete valuation ring of $\raisebox{1ex}{$K$}\!\left/ \!\raisebox{-1ex}{$\kappa $}\right.$. E. Kunz showed that if $\omega$ is a closed differential form and ${\nu }_R\left(\omega \right)\ge -1$, then $re{s}_{R,\underset{¯}{t}}\left(\omega \right)$ does not depend on the choice of parameter $\underset{¯}{t}=\left\{t_1,t_2,\cdots ,t_n\right\}$.

In this paper, we investigate $re{s}_{R,\underset{¯}{t}}\left(\omega \right)$ in the case where ${\nu }_R\left(\omega \right)\ge -p^m+1$ for $\omega \in Z_m$.

Loading vectors and loading coefficients of the parameters of a $3^m$ factorial design and the characteristic vector of its information matrix are introduced. Specific properties of an orthogonal design derived from three-symbol orthogonal array of strength two are discussed. Orthogonal $18$-run $3^4$ factorial designs obtained respectively from the representatives of twelve isomorphic classes are reviewed and two designs among them are recommended for use from the practical point of view.

Friedland and Hsiung [1] proved an analogue of F. Schur’s theorem concerning the holomorphic sectional curvature of some almost Hermitian manifolds called almost Hermitian $L$-manifolds, of which Kählerian manifolds are special ones. Recently, Hsiung and Xiong [3] gave a classification of almost Hermitian manifolds and extended the above work of Friedland and Hsiung to a new class of almost Hermitian manifolds called the class of almost $C$ Hermitian manifolds.

In this paper we shall further extend the above work of Hsiung and Xiong by studying the general sectional, the holomorphic sectional and the holomorphic bisectional curvatures of almost Hermitian manifolds of all classes, together with some relationship among the three types of sectional curvatures.

Let $P$ be a finite set of points in the 3-dimensional Euclidean space ${ℝ}^3$ in general position. For $x_0,x_1,x_{2,}{x}_3\in P$, let $H^{+}\left(x_0;x_1,x_{2,}{x}_3\right)$ (resp. $res.H^{-}\left(x_0;x_1,x_{2,}{x}_3\right)$) denote the open half space containing $x_0$ (resp. not containing $x_0$) and bounded by the plane containing $x_{1,}{x}_{2,}{x}_3$. Further let

In this paper, we show the following statement: if $\left|P\right|\ge 4$, and if ${\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4$ are integers with ${\kappa }_1+{\kappa }_2+{\kappa }_3+{\kappa }_4=\left|P\right|-4,0\le {\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4\le \frac{\left|P\right|-2}{2}$ and ${\kappa }_1+{\kappa }_2\le \frac{\left|P\right|-2}{2}$, then for any $p_1,p_2\in P\left(p_1\ne p_2\right)$, there exist $q_1,q_2\in P$ such that the convex hull of $\left\{p_1,p_{2,}{q}_1,q_2\right\}$ is a 3-simplex (tetrahedron) containing no point of $P$ in its interior and such that

$$\begin{array}{l}|P(p_1;p_2,q_1,q_2)\le k_1\le P\cap H^{-}(p_1;p_2,q_{1,}{q}_2)|,\\ |P(p_2;p_1,q_1,q_2)\le k_2\le P\cap H^{-}(p_2;p_1,q_{1,}{q}_2)|,\\ |P(q_1;q_2,p_1,p_2)\le k_3\le P\cap H^{-}(q_1;q_2,p_{1,}{p}_2)|,\\ |P(q_2;q_1,p_1,p_2)\le k_4\le P\cap H^{-}(q_2;q_1,p_{1,}{p}_2)|.\end{array}$$

Recently, it has been revealed that the semigroups satisfying Gaussian estimates inherit some of the nice properties enjoyed by the Gaussian semigroup itself. Arendt [1] gives a result in this direction, by proving the invariance of the spectrum of the generators of consistent $C_0$-semigroups with Gaussian estimates. In this paper, we generalize this result to the semigroups estimated by the one generated by the fractional power .