The Poincaré disk is a typical example of a (real inner product) gyrovector spaces. In this paper, we show that the Poincaré disk cannot have a complex scalar multiplication in a natural sense.

A modified subformula property for the modal logic KG1 is shown, where KG1 is the smallest normal modal logic K with the additional axiom $♢\square A\supset \square ♢A$ called G1, and it is characterized by the convergent frames. The finite model property and decidability of this logic are easy corollaries.

We derive an explicit formula for the harmonic representation of $SU(p,q)$ in the Fock space from the holomorphic representations of the semi-direct product of the $(2p+2q+1)$-dimensional Heisenberg group by $SU(p,q)$.

Let $k$ be an algebraically closed field of positive characteristic $p$. In this article, we classify representations of ${\mathbb{G}}_a⋊{\mathbb{G}}_m$ into $\text{SL}(3,k)$, and thereby we classify fundamental representations of ${\mathbb{G}}_a$ into $\text{SL}(3,k)$.