We study the branching rule for the restriction from a complex simple Lie group $G$ to the normalizer of a maximal torus of $G$. We show that the problem is reduced to the determination of the Weyl group module structures induced on the zero weight spaces of representations of semisimple Lie groups. The concrete formulas are obtained for $SL$($n$, C) in terms of generalized q-binomial coeffcients and Schur functions.

The branching rule for $SO(n+3)/SO(3) \times SO(n)$ is discussed. An effective bound for the stability in the branching is given.

Green function of 2-point simple-type self-adjoint boundary value problem for 4-th order linear ordinary differential equation, which represents bending of a beam with the boundary condition as clamped, Dirichlet, Neumann and free. The construction of Green function needs the symmetric orthogonalization method in some cases. Green function is the reproducing kernel for suitable set of Hilbert space and inner product. As an application, the best constants of the corresponding Sobolev inequalities are expressed as the maximum of the diagonal values of Green function.

We represent the convergence rates of approximate sums of the areas of surfaces of revolution as limits of their expanded error terms and estimate them. In the case of convex surfaces of revolution we represent the convergence rates of them by the integral of certain functions.

Let $A \in M_n(\mathbf{C})$, $n \ge 2$ be the matrix which has at least one real eigenvalue $\alpha \in (0, 1)$. If the matrix equation \begin{equation} A^x + A^y + A^z = A^w \tag{1} \end{equation} is satisfied in positive integers $x$, $y$, $z$, $w$, then $\max \{x-w, y-w, z-w\} \ge 1$. If suppose that the matrix $A$ has at least one real eigenvalue $\alpha > \sqrt{2}$ and the equation (1) is satisfied in positive integers $x$, $y$, $z$ and $w$, then $\max \{x-w, y-w, z-w\} = -1$. Moveover, we investigate the solvability of the matrix equations (1) and \begin{equation} A^x + A^y = A^z \tag{2} \end{equation} for the non-negative real $n \times n$ matrices, where $|\det A| > 1$, in positive integers $x$, $y$, $z$, $w$ for (1) and $x$, $y$, $z$ for (2). Using the wellknown theorem of Perron-Frobenius we obtain some informations concerning solvability these equations.

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study quasi-Einstein spacetimes. Some basic geometric properties of such a spacetime are obtained. The applications of quasi-Einstein spacetimes in general relativity and cosmology are investigated. Finally, the existence of such spacetimes are ensured by several interesting examples.