For every sense-preserving self-homeomorphism of the real axis, Hubbard constructed an extension that is a self-homeomorphism of the upper half-plane by triangulation. It is natural to ask if such extensions of quasisymmetric homeomorphisms of the real axis are all quasiconformal. Furthermore, for what sense-preserving self- homeomorphisms are such extensions David mappings? In this article, a sufficient and necessary condition for such extensions to be quasiconformal and a sufficient condition for such extensions to be David mappings are given.

Let $R$ and $S$ be rings, and let ${}_R{\omega }_S$ be a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsion-free modules and a subclass of the class of $\omega$-adstatic modules. Also, we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of $Hom(\omega ,M)$. By investigating the properties of the Bass injective dimension of modules (resp., complexes), we get some equivalent characterizations of semitilting modules (resp., Gorenstein Artin algebras). Finally, we obtain a dual version of the Auslander–Bridger approximation theorem. As a consequence, we get some equivalent characterizations of Auslander $n$-Gorenstein Artin algebras.

We discuss the concept of Cremona contractible plane curves, with a historical account on the development of this subject. Then we classify Cremona contractible unions of $d\ge 12$ lines in the plane.

In this article, we consider Trudinger’s inequality and continuity for Riesz potentials of functions in Orlicz spaces of two variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.

Une variété de Fano faible est une variété admettant un diviseur anticanonique gros et numériquement effectif. On donne une caractérisation des volumes de Fano faibles obtenus par éclatement d’une courbe de ${\mathbf{P}}^3$. Le résultat est optimal.

We characterize smooth curves in ${\mathbf{P}}^3$ whose blowup produces a threefold with big and nef anticanonical divisor.

We study moduli spaces of sheaves over nonprojective K3 surfaces. More precisely, let $\omega$ be a Kähler class on a K3 surface $S$, let $r\ge 2$ be an integer, and let $v=(r,\xi ,a)$ be a Mukai vector on $S$. We show that if the moduli space $M$ of ${\mu }_{\omega }$-stable vector bundles with associated Mukai vector $v$ is compact, then $M$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. Moreover, we show that there is a Hodge isometry between $v^{\perp }$ and $H^2(M,\mathbb{Z})$ and that $M$ is projective if and only if $S$ is projective.

Our aim in this article is to deal with boundary limits of monotone Sobolev functions in Musielak–Orlicz spaces on uniform domains in a metric space.

A planar open Riemann surface $R$ admits the Schiffer span $s(R,\zeta )$ to a point $\zeta \in R$. M. Shiba showed that an open Riemann surface $R$ of genus one admits the hyperbolic span ${\sigma }_H\left(R\right)$. We establish the variation formulas of ${\sigma }_H\left(t\right):={\sigma }_H\left(R\right(t\left)\right)$ for the deforming open Riemann surface $R\left(t\right)$ of genus one with complex parameter $t$ in a disk $\Delta$ of center $0$, and we show that if the total space $\mathcal{R}={\bigcup }_{t\in \Delta }(t,R(t\left)\right)$ is a two-dimensional Stein manifold, then ${\sigma }_H\left(t\right)$ is subharmonic on $\Delta$. In particular, ${\sigma }_H\left(t\right)$ is harmonic on $\Delta$ if and only if $\mathcal{R}$ is biholomorphic to the product $\Delta \times R\left(0\right)$.

We give the characterization of the differentiability and nondifferentiability points of a generalization of the Riesz–Nágy–Takács (RNT) singular function, namely, the generalized RNT (GRNT) singular function. A particular characterization generalizes recent multifractal and metric-number-theoretical results associated with the RNT singular function. Furthermore, we compute the moments of the GRNT singular function.

The typical chaotic maps $f\left(x\right)=4x(1-x)$ and $g\left(z\right)=z^2-2$ are well known. Veselov generalized these maps. We consider a class of maps $P_{A_3}^d$ of those generalized maps, view them as holomorphic endomorphisms of ${\mathbb{P}}^3\left(\mathbb{C}\right)$, and make use of methods of complex dynamics in higher dimension developed by Bedford, Fornaess, Jonsson, and Sibony. We determine Julia sets $J_1,J_2,J_3,J_{\Pi }$ and the global forms of external rays. Then we have a foliation of the Julia set $J_2$ formed by stable disks that are composed of external rays.

We also show some relations between those maps and catastrophe theory. The set of the critical values of each map restricted to a real three-dimensional subspace decomposes into a tangent developable of an astroid in space and two real curves. They coincide with a cross section of the set obtained by Poston and Stewart where binary quartic forms are degenerate. The tangent developable encloses the Julia set $J_3$ and joins to a Möbius strip, which is the Julia set $J_{\Pi }$ in the plane at infinity in ${\mathbb{P}}^3\left(\mathbb{C}\right)$. Rulings of the Möbius strip correspond to rulings of the surface of $J_3$ by external rays.