Showing papers in "Tsukuba journal of mathematics in 2009"
TL;DR: The notion of quasi-Einstein spacetimes arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasiumbilical hypersurfaces.
Abstract: 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.
52 citations
TL;DR: In this article, the authors consider the initial-boundary value problems for the reaction-diffusion systems with inhomogeneous terms in cones and show the nonexistence of global solutions of the problems in time.
Abstract: We consider initial-boundary value problems for the reaction-diffusion systems with inhomogeneous terms in cones. In this paper we show the nonexistence of global solutions of the problems in time.
15 citations
TL;DR: 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.
Abstract: 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.
12 citations
TL;DR: In this paper, the harmonic volume of the Fermat sextic was computed using a special algebraic cycle in the Jacobian variety of the Sextic and proved that it is not algebraically equivalent to zero.
Abstract: We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
9 citations
TL;DR: In this paper, the authors studied real hypersurfaces in a complex space form whose structure Jacobi operator (R_\xi=R(\cdot,\xi)\xi$ is parallel.
Abstract: Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(\phi, \xi, \eta, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_\xi=R(\cdot,\xi)\xi$ is $\xi$-parallel. In particular, we prove that the condition $
abla_{\xi} R_{\xi}=0$ characterizes the homogeneous real hypersurfaces of type $A$ in a complex projective space or a complex hyperbolic space when $R_{\xi}\phi S=S\phi R_{\xi}$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$.
8 citations
TL;DR: In this article, Suzuki et al. considered a product interpretation of a Presburger arithmetic with a new predicate and showed that it admits quantifier elimination in the language of the abelian groups.
Abstract: Let $\Lg = \{+,-,0\}$ be the language of the abelian groups, $L$ an expansion of $\Lg(<)$ by relations and constants, and $\Lmod = \Lg \cup \{\equiv_n\}_{n \geq 2}$ where each $\equiv_n$ is defined as follows: $x \equiv_n y$ if and only if $n|x-y$. Let $H$ be a structure for $L$ such that $H|\Lg(<)$ is a totally ordered abelian group and $K$ a totally ordered abelian group. We consider a product interpretation of $H \times K$ with a new predicate $I$ for $\{0\}\times K$ defined by N.~Suzuki \cite{Sz}. Suppose that $H$ admits quantifier elimination in $L$. 1. If $K$ is a Presburger arithmetic with smallest positive element $1_K$ then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, 1) \cup \Lmod$ with $1^G = (0^H, 1_K)$.2. If $K$ is dense regular and $K/nK$ is finite for every integer $n \geq 2$ then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ for some set $D$ of constant symbols where $G \models I(d)$ for each $d \in D$.3. If $K$ admits quantifier elimination in $\Lmod(<, D)$ for some set $D$ of constant symbols then the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ unless $K$ is dense regular with $K/nK$ being infinite for some $n$. Conversely, if the product interpretation $G$ of $H \times K$ with a new predicate $I$ admits quantifier elimination in $L(I, D) \cup \Lmod$ for some set $D$ of constant symbols such that $G \models I(d)$ for each $d \in D$ then $H$ admits quantifier elimination in $L \cup \Lmod$, and $K$ admits quantifier elimination in $\Lmod(<, D)$. We also discuss the axiomatization of the theory of the product interpretation of $H \times K$. %For some set $C$ of constants in $K$.
5 citations
TL;DR: In this article, the authors gave conditions to have only finitely many orbits for prehomogeneous vector spaces of type D_4-type with cost O(n 2 ) for the complex number field.
Abstract: In the present paper, we give conditions to have only finitely many orbits for prehomogeneous vector spaces of $D_4$-type. This paper completes the classification of finite prehomogeneous vector spaces of type $(G \times SL_n, \rho \otimes \Lambda_1)$ with $n \geq 2$. We consider everything over the complex number field $\mathbb{C}$.
2 citations
2 citations
TL;DR: In this article, the authors studied the normalizer subgroup of a maximal torus of the classical group of type C, and obtained a formula of the irreducible characters of this subgroup.
Abstract: We study representations of the normalizer subgroup $N$ of a maximal torus of the classical group of type C, $Sp(n)$ We obtain a formula of the irreducible characters of $N$, and give the branching rule from $Sp(n)$ to $N$
1 citations
1 citations
TL;DR: In this article, the authors investigated the solvability of the matrix equations (1) and (2) for non-negative real matrices with at least one real eigenvalue in positive integers.
Abstract: 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
TL;DR: In this article, the branching rule for $SO(n+3)/SO(3) \times SO(n)$ is discussed and an effective bound for the stability in the branching is given.
Abstract: 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.
TL;DR: In this article, the branching rule for the restriction from a simple Lie group to the normalizer of a maximal torus was studied and the problem was reduced to the determination of the Weyl group module structures induced on the zero weight spaces of representations of semisimple Lie groups.
Abstract: 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.
TL;DR: In this paper, the authors classify simple prehomogeneous vector spaces of type (GL{1}^{l}times G{1}\times \cdots \times G_{k},
Abstract: In this paper, we classify $k$-simple prehomogeneous vector spaces of type $(GL_{1}^{l}\times G_{1}\times \cdots \times G_{k},\rho _{1}^{(1)}\otimes \cdots \otimes \rho _{k}^{(1)}+ \cdots + \rho_{1}^{(l)}\otimes \cdots \otimes \rho _{k}^{(l)})$ where for any $i,j$, each $\rho _{j}^{(i)}$ is a nontrivial irreducible representation of a simple algebraic group $G_{j}$(i.e., $\rho _{j}^{(i)}
eq 1$) with $k\ge 3$ and $l\ge 2$ under full scalar multiplications. We consider everything over the complex number field $\mathbb {C}$.