Quaternionic numerical range is not always a convex set. In this note, an explicit criterion is given for the convexity of quaternionic numerical range.

In this note we determine which dihedral subgroups of $\text{G}{{\text{L}}_{g}}\,(\mathbb{C})$ can be realized by group actions on Riemann surfaces of genus $g>1$.

Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M. Hartl identified the subgroup $G\,\cap \,(1+\,{{\Delta }^{3}}\,(G)\,+\,\Delta (G)\Delta (H))$ of $G$. In this note we give an independent proof of the result of Hartl, and we identify two subgroups $G\,\cap \,(1\,+\,\Delta (H)\Delta (G)\Delta (H)\,+\,\Delta (\left[ H,\,G \right]\Delta (H)),\,G\,\cap \,(1\,+\,{{\Delta }^{2}}\,(G)\Delta (H)\,+\,\Delta (K)\Delta (H))$ of $G$ for some subgroup $K$ of $G$ containing $[H,G]$.

Let ${{M}_{n}}(F)$ be the algebra of $n\times n$ matrices over a field $F$ of characteristic $p>2$ and let $*$ be an involution on ${{M}_{n}}(F)$. If ${{s}_{1}},...,{{s}_{r}}$ are symmetric variables we determine the smallest $r$ such that the polynomial

$${{P}_{r}}({{S}_{1}},...,{{S}_{r}})\,=\,\sum\limits_{\sigma \in {{S}_{r}}}{{{S}_{\sigma (1)}}...{{S}_{\sigma (r)}}}$$

is a $*$-polynomial identity of ${{M}_{n}}(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial

$${{C}_{r}}\left( {{k}_{1}},...,{{k}_{r,}}{{{{k}'}}_{1}},...,{{{{k}'}}_{r}} \right)=\sum\limits_{\sigma ,\tau \in {{S}_{r}}}{{{k}_{\sigma \left( 1 \right)}}{{{{k}'}}_{\tau \left( 1 \right)}}\cdot \cdot \cdot {{k}_{\sigma \left( r \right)}}{{{{k}'}}_{\tau \left( r \right)}}}$$

where ${{k}_{1}},...,{{k}_{r}},k_{1}^{'},...,k_{r}^{'}$ are skew variables under the transpose involution.

If $f({{x}_{1}},...,{{x}_{k}})$ is a polynomial with complex coefficients, the Mahler measure of $f$ , $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus ${{\mathbb{T}}^{k}}$. We construct a sequence of approximations ${{M}_{n}}\,(f)$ which satisfy $-d{{2}^{-n}}\,\log \,2\,+\,\log \,{{M}_{n}}(f)\,\le \,\log \,M(f)\,\le \,\log \,{{M}_{n}}(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since ${{M}_{n}}\,(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.