We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of ${{t}^{-5/6}}$. This rate is due to competition between surface tension and gravitation at $O(1)$ wave numbers and is connected to the fact that, in the presence of surface tension, there is a so-called “slowest wave”. Additionally, we combine our dispersive estimates with ${{L}^{2}}$ type energy bounds to prove a family of Strichartz estimates.

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.

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.

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.