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 $N$.

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.

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 $M$.

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 $\mathbb{C}$.

The box complex ${\sf B}(G)$ of a graph $G$ is a simplicial $\mathbf{Z}_2$-complex defined by J. Matoušek and G.M. Ziegler in \cite{MZ04}. They proved that $\chi (G)\geq \text{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| )+2$, where $\chi (G)$ is the chromatic number of $G$ and $\text{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| )$ is the $\mathbf{Z}_2$-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 ${\sf B}(G\cup H)={\sf B}(G)\cup {\sf B}(H)$ and ${\sf B}(G\cap H)={\sf B}(G)\cap {\sf B}(H)$ hold. Moreover, under that condition, we show $$\rm{max} \{\chi (G), \chi (H)\}\leq \chi (G\cup H)\leq \max \{\chi (G)+l_H, \chi (H)\},$$

where $l_H$ is the number defined in Definition 3.8. Also we prove $$\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(G\cup H)\| )=\max \{\,\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| ),\, \rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(H)\| )\,\}$$

if $\max \{\,\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| ),\,\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(H)\| )\,\}\geq 1$.

The complex $\mathsf{B}(G)$ of a graph $G$ contains a 1-dimensional free $\mathbf{Z}_2$-subcomplex $\overline{G}$ of ${\sf B}(G)$, defined in [2]. As a supplement to [2], we show that for a connected graph $G$, $\mathsf {B}(G)$ is disconnected if and only if $\overline{G}$ is disconnected if and only if $G$ contains no cycles of odd length, or equivalently, $G$ is bipartite.

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 $K$.

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.

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.

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 $\mathbb {C}$.