Given a positive integer $n$, a finite field ${{\mathbb{F}}_{q}}$ of $q$ elements ($q$ odd), and a non-degenerate symmetric bilinear form $B$ on $\mathbb{F}_{q}^{n}$, we determine the largest possible cardinality of pairwise $B$-orthogonal subsets $\varepsilon \,\subseteq \,\mathbb{F}_{q}^{n}$, that is, for any two vectors $x,\,y\,\in \,\varepsilon $, one has $B(x,\,y)\,=\,0$.

We investigate the solutions of refinement equations of the form

$$\phi (x)\,=\,\sum\limits_{\alpha \in {{\mathbb{Z}}^{S}}}{a(\alpha )\,}\phi (Mx\,-\,\alpha ),$$

where the function $\phi $ is in ${{L}_{p}}({{\mathbb{R}}^{s}})(1\,\le \,p\,\le \,\infty )$, $a$ is an infinitely supported sequence on ${{\mathbb{Z}}^{s}}$ called a refinement mask, and $M$ is an $s\,\times \,s$ integer matrix such that ${{\lim }_{n\to \infty }}\,{{M}^{-n}}\,=\,0$. Associated with the mask $a$ and $M$ is a linear operator ${{\text{Q}}_{a,M}}$ defined on ${{L}_{p}}({{\mathbb{R}}^{s}})$ by ${{\text{Q}}_{a,M}}{{\phi }_{0}}\,:=\,{{\sum }_{\alpha \in {{\mathbb{Z}}^{s}}}}\,a(\alpha ){{\phi }_{0}}(M\,\cdot \,-\alpha )$. Main results of this paper are related to the convergence rates of ${{(\text{Q}_{a,M}^{n}{{\phi }_{o}})}_{n=1,2,\ldots }}$ in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function ${{\phi }_{0}}$, $\text{Q}_{a,M}^{n}{{\phi }_{0}}$ converges in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with an exponential rate.

We consider the diophantine equation ${{x}^{2}}\,+\,{{y}^{6}}\,=\,{{z}^{e}},\,e\,\le \,4$. We show that, when $e$ is a multiple of 4 or 6, this equation has no solutions in positive integers with $x$ and $y$ relatively prime. As a corollary, we show that there exists no primitive Pythagorean triangle one of whose leglengths is a perfect cube, while the hypotenuse length is an integer square.

While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces ${{B}^{\alpha }}$ with $\alpha \,\ne \,1$ are the ones induced by a disc automorphism.