The $CR$ analogue of B.-Y. Chen's conjecture on pseudo biharmonic maps will be shown. Pseudo biharmonic, but not pseudo harmonic, isometric immersions with pseudo parallel pseudo mean curvature vector fields, will be characterized.

Let $G$ be a finite group, and $\mathscr{C}$ a $G$-abelian category. We prove that the skew group category $\mathscr{C}(G)$ is an abelian category under the condition that the order $|G|$ is invertible in $\mathscr{C}$. When the order $|G|$ is not invertible in $\mathscr{C}$, an example is given to show that $\mathscr{C}(G)$ is not an abelian category.

We give integral representations of positive and negative definite functions defined on an interval in a certain subsemigroup of the semigroup of rational numbers.

Let $R$ be a commutative ring, and let $A$ be an associative $R$-algebra possessing an $R$-free basis $B$. In this paper, we introduce a homology $H_{n}(A,B)$ associated to a pair $(A,B)$ under suitable hypotheses. It depends on not only $A$ itself but also a choice of $B$. In order to define $H_{n}(A,B)$, we make use of a certain submodule of the $(n+1)$-fold tensor product of $A$. We develop a general theory of $H_{n}(A,B)$. Various examples of a pair $(A,B)$ and $H_{n}(A,B)$ are also provided.

In this paper, we consider the initial value problem for nonlinear wave equation with weighted nonlinear terms in one space dimension. Kubo & Osaka & Yazici [4] studied global solvability of the problem under different conditions on the nonlinearity and initial data, together with an upper bound of the lifespan for the problem. The aim of this paper is to improve the upper bound of the lifespan and to derive its lower bound which shows the optimality of our new upper bound.