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 use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

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.

Let $C$ be a hyperelliptic curve given by the equation ${{y}^{2}}\,=\,f(x)$ for $f\,\in \,\mathbb{Z}[x]$ without multiple roots. We say that points ${{P}_{i}}\,=\,({{x}_{i}},\,{{y}_{i}})\,\in \,C(\mathbb{Q})$ for $i\,=\,1,\,2,\,\ldots ,\,m$ are in arithmetic progression if the numbers ${{x}_{i}}$ for $i\,=\,1,\,2,\,\ldots ,\,m$ are in arithmetic progression.

In this paper we show that there exists a polynomial $k\,\in \,\mathbb{Z}[t]$ with the property that on the elliptic curve $\varepsilon \prime :{{y}^{2}}={{x}^{3}}+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\varepsilon \prime $. In particular this result generalizes earlier results of Lee and Vélez. We also show that if $n\,\in \,\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves ${{y}^{2}}\,=\,{{x}^{n}}\,+\,k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves ${{y}^{2}}\,=\,{{x}^{n}}\,+\,k$ there are six rational points in arithmetic progression.

Let $G$ be an arbitrary finite abelian group. We describe all possible $G$-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

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.

In this paper, some criteria for the existence of positive solutions of a class of systems of impulsive dynamic equations on time scales are obtained by using a fixed point theorem in cones.