We show that every countable strict directed system of finitedimensional Lie groups has a direct limit in the category of smooth Lie groups modelled on sequentially complete, locally convex spaces. Similar results are obtained for countable directed systems of finite-dimensional manifolds, and for countable directed systems of finite-dimensional Lie groups and manifolds over totally disconnected local fields. An uncountable strict directed system of finite-dimensional Lie groups has a direct limit in the category of Lie groups in the sense of convenient differential calculus, provided certain technical hypotheses are satisfied.

For the family $\mathscr{F}(K)$ of $K$-quasiconformal mappings $f$ from $\mathbb{\bar{C}} = \{|z|\leqslant +\infty \}$ onto $\mathbb{C}$ such that $f(\mathbb{R}) = \mathbb{R}$ and $f(x) = x$ for $x=-1$, $0$, $\infty$, the supremum $\lambda (K, t)$ and the infimum $\nu (K, t)$ of $f(t)$ for $f$ ranging over $\mathscr{F}(K)$ with $t \in \mathbb{R}$ fixed are studied. They are expressed by the inverse $\mu ^{-1}$ of the function $\mu (r)$, the modulus of the bounded, doubly-connected domain with the unit circle and the real interval $[0, r]$, $0 < r < 1$, as the boundary. Among a number of results obtained, asymptotic behaviors of $X(K, t)(X = \lambda , \nu )$ as $t \to \pm \infty$ for a fixed $K$ and as $K \to +\infty$ for a fixed $t$ are considered.

Let $S$ be a rank one log del Pezzo surface of index two and $S^{0}$ the smooth part of $S$. In this paper we determine the singularity type of $S$, in a way different from Alekseev and Nikulin [1]. Moreover, we calculate the fundamental group of $S^{0}$ and prove that $S$ contains the affine plane as a Zariski open subset if and only if $\pi _{1}(S^{0})=(1)$.

Let $p$ be a prime, $\mathbb{Q}_{p}$ the field of $p$-adic numbers and $\mathbb{\bar{Q}}_{p}$ a fixed algebraic closure of $\mathbb{Q}_{p}$. $B_{dR}^{+}$ is the ring of $p$-adic periods of algebraic varieties over $p$-adic fields introduced by Fontaine. For each n one defines a canonical valuation $w_{n}$ on $\mathbb{\bar{Q}}_{p}$ such that $B_{dR}^{+}/I^{n}$ becomes the completion of $\mathbb{\bar{Q}}_{p}$ with respect to $w_{n}$, where $I$ is the maximal ideal of $B_{dR}^{+}$. An element $\alpha \in \mathbb{\bar{Q}}_{p}^{*}$ is said to be good at level $n$ if $w_{n}(\alpha ) = v(\alpha )$ where $v$ denotes the $p$-adic valuation on $\mathbb{\bar{Q}}_{p}$. The set $\mathcal{G}_{n}$ of good elements at level n is a subgroup of $\mathbb{\bar{Q}}_{p}^{*}$. We prove that each quotient group $\mathbb{\bar{Q}}_{p}^{*}/\mathcal{G}_{n}$ is a torsion group and that each quotient $\mathcal{G}_{1}/\mathcal{G}_{n}$ is a $p$-group. We also show that a certain sequence of metric invariants $\{ l_{n}(Z)\}_{n\in \mathbb{N}}$ associated to an element $Z \in B_{dR}^{+}$, is constant.

In this paper, we consider an inverse scattering problem for an obstacle $D \subset \mathcal{R}^{3}$ with Robin boundary condition. A reconstruction procedure for identifying both the shape of the obstacle and the boundary impedance from the far-field pattern is proposed. Our method is to transform the far-field patterns into the near-field patterns, then construct an indicator function from which we can determine the boundary shape. Having known the shape of $D$, the boundary impedance is recovered by moment method for the known boundary.

We shall give a concrete description of minimal algebraic surfaces $X$’s defined over $\mathbb{C}$ of general type with the first chern number 3, the geometric genus 1 and the irregularity 0, which have non-trivial 3-torsion divisors. Namely, we shall show that the fundamental group is isomorphic to $\mathbb{Z}/3$, and that the canonical model of the universal cover is a complete intersection in $\mathbb{P}^{4}$ of type (3, 3).

Kra [Kr] introduced a distance $d_{K}$ on every hyperbolic Riemann surface $R$ by means of Teichmüller shift mappings. Recently Gardiner and Lakic [GL2] defined a metric density $\lambda$, the Teichmüller density, on such a surface. The paper deals with some basic properties of the Teichmüller density $\lambda$ and the distance $d_{K}$, giving some close relation between them. Particularly, it is shown that the distance function $d_{K} : R \times R \to \mathbb{R}$ is continuously differentiable off the diagonal, and the Teichmüller density $\lambda$ is precisely the metric density of the infinitesimal form of the distance $d_{K}$ and it is continuous on the whole surface $R$. Some related topics will also be discussed.