We consider families of cubic fields introduced by Ishida. We find integral bases and the fundamental units for these families.

Let $f(x)=x^3+3x+a^3(a\in \mathbf{Z})$ be a cubic polynomial and $\theta$ be the real root of $f(x)$. We consider the unit group of $\mathbf{Q}(\theta )$. We show that $\eta =1-a^2-\alpha \theta$ is a fundamental unit of $\mathbf{Q}(\theta )$ under certain conditions. And we consider the 3-class group of $\mathbf{Q}(\theta )$.

Existence of a nontrivial solution to a semilinear elliptic equation is established by a variational method. We consider the weak solution to $-\mathrm{\Delta }u=h(x,u)$, where $h(x,u)$ is (asymptotically linear in $u$ both as $u\to 0$ and $u\to \infty$. The proof is based on local linking theory, $(PS{)}^{*}$ condition and approximation by finite dimensional subspaces for the existence of a nontrivial critical point of a $C^1$-class functional.

We introduce and study here the notion of $f$-pluriharmonicity, as the extension of pluriharmonicity from the context of the almost Hermitian manifolds, to the manifolds endowed with $f$-structures, which are defined by K. Yano in [12]. Then we relate $f$-pluriharmonicity with $\pm f$-holomorphicity and $f-(1,1)$–geodesicity. We generalize a result obtained by S. Udagawa in [11], we give some applications by using the complex sectional curvature defined by T. Siu in [9] and we construct some examples.

The first purpose of this paper is to give a smart proof of the Morse index theorem for squared distance function of submanifolds in s a symmetric space. The second purpose is to classify focal points into strong ones and weak ones and to give a class of submanifolds in symmetric spaces all of whose focal points (other than conjugate points) are strong ones. The third purpose is to construct examples of submanifolds in symmetric spaces admitting weak ones.

We study a complex equiaffine immersion of general codimension by regarding a frame of the complex determinant bundle of a complex vector bundle as a complex volume form. Some results on a complex affine hypersurface with volume form, especially a complex equiaffine hypersurface, are generalized to the case of general codimension. Especially, we obtain the fundamental theorems for a complex equiaffine immersion to a complex affine space of general codimension.

For a graph $G$, a subset $S$ of $V(G)$ is called a shredder if $G-S$ consists of three or more components. We show that if $G$ is a 6-connected graph of order at least 325, then the number of shredders of cardinality 6 of $G$ is less than or equal to $(2|V(G)|-9)/3$.

In this paper, we prove the following result:

Let $k$ be an integer with $k\ge 5$. Let $C$ be a cycle in a graph $G$, and let $H$ be a component of $G-C$. Suppose that $C$ is locally longest with respect to $H$, and $H$ is locally $k$-connected to $C$, $\left|V(H)\right|\ge k-1$, $\delta (H)\ge \lfloor (k-1)/2\rfloor$, and $H$ is 3-connected. Let $r=({\sum }_{x\in V(H)}{\mathrm{deg}}_G(x))/|V(H)|$. Then $l(C)\ge k(r+2-k)$, with equality only if $r$ is an integer and either $H$ is a complete graph of order $r+1-k$ and every vertex of $H$ has the same $k$ neighbours on $C$, or $H$ is a complete graph of order $k-1$ and every vertex of $H$ has the same $r+2-k$ neighbours on $C$.