Let $f$ be a transcendental meromorphic function in the complex plane $\mathbf{C}$, and a be a nonzero complex number. We give quantitative estimates for the characteristic function $T(r,f)$ in terms of $N(r,1/( f^1(f^{(k)})^n-a))$, for integers $k$, $l$, $n$ greater than 1. We conclude that $f^1(f^{(k)})^n$ assumes every nonzero finite value infinitely often.

We give upper bounds of the Matveev complexities of two-bridge link complements by constructing their spines explicitly. In particular, we determine the complexities for an infinite sequence of two-bridge links corresponding to the continued fractions of the form $[2,1,\dots, 1,2]$. We also give upper bounds for the 3-manifolds obtained as meridian-cyclic branched coverings of the 3-sphere along two-bridge links.

By using an elliptic analogue of the Drinfeld coproduct, we construct the level-$(k+1)$ representation of the elliptic quantum group $U_{q,p}(\widehat{\mathfrak{sl}}_2)$ from the level-1 highest weight representation. The quantum Z-algebra of level-$(k+1)$ is realized. We also find the elliptic analogue of the condition of integrability for higher level modules constructed by the Drinfeld coproduct. This also enables us to express $\Delta^k(e(z))\Delta^k(e(zq^2))\dots\Delta^k(e(zq^{2(N-1)}))$ and $\Delta^k(f(z))\Delta^k(f(zq^2))\Delta^k(f(zq^{-2}))\dots\Delta^k(f(zq^{-2(N-1)}))$ as vertex operators of the level-$(k+1)$ bosons.

This note gives a new sufficient condition of the total energy decay for the solutions of the initial-boundary value problems to the dissipative wave equations in exterior domains with non-compactly supported initial data. That condition provides an example of the damping terms of the dissipative wave equations with the total energy decay which has a smaller amplitude than those of all examples derived from a sufficient condition in Mochizuki and Nakazawa [Publ. Res. Inst. Math. Sci. 32 (1996), 401–414].

We work over an algebraically closed field $k$ of positive characteristic $p$. Let $q$ be a power of $p$. Let $A$ be an $(n+1)\times(n+1)$ matrix with coefficients $a_{ij}$ in $k$, and let $X_A$ be a hypersurface of degree $q + 1$ in the projective space $\mathbf{P}^n$ defined by $\sum a_{ij}x_i x^q_j=0$. It is well-known that if the rank of $A$ is $n + 1$, the hypersurface $X_A$ is projectively isomorphic to the Fermat hypersuface of degree $q + 1$. We investigate the hypersurfaces $X_A$ when the rank of $A$ is $n$, and determine their projective isomorphism classes.

The once-punctured torus and the once-punctured Klein bottle are topologically commensurable, in the sense that both of them are doubly covered by the twice-punctured torus. In this paper, we give a condition for a faithful type-preserving $\mathrm{PSL}(2,\mathbf{C})$-representation of the fundamental group of the once-punctured Klein bottle to be ‘‘commensurable’’ with that of the once-punctured torus. We also show that such a pair of $\mathrm{PSL}(2,\mathbf{C})$-representations extend to a representation of the fundamental group of a common quotient orbifold. Finally, we give an application to the study of the Ford domains.