## Recursos de colección

1. #### A classification of reductive prehomogeneous vector spaces with trivial representation free

Ouchi, Masaya
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., $\rho _{j}^{(i)}\neq 1$) with $k\ge 3$ and $l\ge 2$ under full scalar multiplications. We consider everything over the complex number field $\mathbb {C}$.

2. #### The lifting of elliptic modular forms to Hilbert modular forms and Petersson inner products

Kojima, Hisashi

3. #### Circular billiards and parallel axiom in convex billiards

Tamura, Shinetsu; Innami, Nobuhiro
Circles will be characterized by some properties of billiard ball trajectories. The theory of parallels and the parallel axiom play important roles in the geometry of the configuration space. Those characterizations are concerned with Bialy's theorem which is a partial answer to Birkhoff's conjecture.

4. #### Nonexistence of Global Solutions in Time for Reaction-Diffusion Systems with Inhomogeneous Terms in Cones

Igarashi, Takefumi; Umeda, Noriaki
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.

5. #### Quantifier Elimination for Lexicographic Products of Ordered Abelian Groups

Ibuka, Shingo; Kikyo, Hirotaka; Tanaka, Hiroshi
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 $L$. [start-list] *1. If $K$ is a Presburger...

6. #### Subcomplexes of box complexes of graphs

Kamibeppu, Akira
The box complex ${\sf B}(G)$ of a graph $G$ is a simplicial $\mathbb{Z}_2$-complex defined by J. Matoušek and G.M. Ziegler in \cite{MZ04}. They proved that $\chi (G)\geq \text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G)\| )+2$, where $\chi (G)$ is the chromatic number of $G$ and $\text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G)\| )$ is the $\mathbb{Z}_2$-index of ${\sf B}(G)$. In this paper, to study topology of box complexes, for the union $G\cup H$ of two graphs $G$ and $H$, we compare ${\sf B}(G\cup H)$ with its subcomplex ${\sf B}(G)\cup {\sf B}(H)$. We give a sufficient condition on $G$ and $H$ so that ${\sf B}(G\cup H)={\sf B}(G)\cup {\sf B}(H)$...

7. #### A characterization of finite prehomogeneous vector spaces of $D_4$-type under various scalar restrictions

Kamiyoshi, Tomohiro
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 $\mathbb{C}$.

8. #### Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form

Ki, U-Hang; Kurihara, Hiroyuki; Takagi, Ryoichi
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 $\nabla_{\xi} R_{\xi}=0$ 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 $M$.

9. #### A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic

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 $N$.