It is known that the modular group $\mathrm{Mod}\left(X\right)$ acts discontinuously (but not freely) on the Teichmüller space $T\left(X\right)$ for a finite type Riemann surface $X$, while it does not necessarily act discontinuously on $T\left(X\right)$ when $X$ is of infinite type. The primary purpose of the paper is to discuss those subgroups of $\mathrm{Mod}\left(X\right)$ acting discontinuously and freely on $T\left(X\right)$ and to discuss the properties of the corresponding quotient complex manifolds as well. Actually, we will discuss some generalized Teichmüller spaces, the Teichmüller spaces for pointed Riemann surfaces and pointed Fuchsian groups, and their modular groups, generalizing and completing some results of Bers [Be1], Kra [Kr1] and Nag ([Na1], [Na3], [Na4]).

Some equivalent characterizations for boundedness of maximal singular integral operators on spaces of homogeneous type are given via certain norm inequalities on John-Strömberg sharp maximal functions and without resorting the boundedness of these operators themselves. As a corollary, the results of Grafakos on Euclidean spaces are generalized to spaces of homogeneous type. Moreover, applications to maximal Monge-Ampère singular integral operators and maximal Nagel-Stein singular integral operators on certain specific smooth manifolds are also presented.

We show that given a finitely presented group $G$ with ${\beta }_1\left(G\right)\geqq 2$ which is a mapping torus ${\mathrm{\Gamma }}_{\theta }$ for $\mathrm{\Gamma }$ a finitely generated group and $\theta$ an automorphism of $\mathrm{\Gamma }$ then if the Alexander polynomial of $G$ is non-constant, we can take ${\beta }_1\left(\mathrm{\Gamma }\right)$ to be arbitrarily large. We give a range of applications and examples, such as any group $G$ with ${\beta }_1\left(G\right)\ge 2$ that is $F_n$-by-$\mathbf{Z}$ for $F_n$ the non-abelian free group of rank $n$ is also $F_m$-by-$\mathbf{Z}$ for infinitely many $m$. We also examine 3-manifold groups where we show that a finitely generated subgroup cannot be conjugate to a proper subgroup of itself.

We establish the universality theorem for the first four symmetric power $L$-functions of automorphic forms and their associated Rankin-Selberg $L$-functions. This generalizes some results of Laurinčikas and Matsumoto and of Matsumoto, respectively.

We study the prototype model of the boundary value problem where $\mathrm{\Omega }\subset {\mathbf{R}}^n$ ($n\ge 2$) is a connected smooth domain, and the exponents $m>1$ and $a,b,c,d\ge 0$ are non-negative numbers. Under appropriate conditions on the exponents $m$, $a$, $b$, $c$ and $d$, and on the domain $\mathrm{\Omega }$, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established.

For any arrangement of hyperplanes in ${\mathbf{CP}}^3$, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. If the soul consists of a set of points (0-simplices) and a set of planes (2-simplices), then the arrangement is called point arrangement. In this paper, we give a sufficient combinatoric condition for two point arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient condition on combinatorics for the point arrangement of hyperplanes whose moduli space is connected.

A bounded Euclidean domain $R$ is said to be a Dirichlet domain if every quasibounded harmonic function on $R$ is represented as a generalized Dirichlet solution on $R$. As a localized version of this, $R$ is said to be locally a Dirichlet domain at a boundary point $y\in \partial R$ if there is a regular domain $U$ containing $y$ such that every quasibounded harmonic function on $U\cap R$ with vanishing boundary values on $\overline{R}\cap \partial U$ is represented as a generalized Dirichlet solution on $U\cap R$. The main purpose of this paper is to show that the following three statements are equivalent by pairs: $R$ is a Dirichlet domain; $R$ is locally a Dirichlet domain at every boundary point $y\in \partial R$; $R$ is locally a Dirichlet domain at every boundary point $y\in \partial R$ except for points in a boundary set of harmonic measure zero. As an application it is shown that if every boundary point of $R$ is graphic except for points in a boundary set of harmonic measure zero, then $R$ is a Dirichlet domain, where a boundary point $y\in \partial R$ is said to be graphic if there are neighborhood $V$ of $y$ and an orthogonal (or polar) coordinate $x=(x\text{'},x^d)$ (or $x=r\xi$) such that $V\cap R$ is represented as one side of a graph of a continuous function $x^d=\varphi (x\text{'})$ (or $r=\varphi \left(\xi \right)$).

We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the $L_q$ framework in a domain of ${\mathbf{R}}^n$ ($n\ge 2$) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: $\nu \cdot u=0$ and on the boundary of the domain with $\alpha ,\beta \ge 0$ and $\alpha +\beta =1$, where $u$ denotes a velocity vector, $p$ a pressure, $T(u,p)$ the stress tensor for the Stokes flow, and $\nu$ the unit outer normal to the boundary of the domain. It presents the slip condition when $\beta =1$ and the non-slip one when $\alpha =1$, respectively.

We introduce the class of concealed generalized canonical algebras and prove that their Auslander-Reiten quiver admits an infinite family of faithful standard stable tubes. Moreover, a new wide class of selfinjective algebras whose Auslander-Reiten quiver admits an infinite family of standard stable tubes is exhibited.

The bicategory of normal functors between W*-categories is shown to be monoidally equivalent to the bicategory of W*-bimodules.

Let $N$ and $P$ be smooth manifolds of dimensions $n$ and $p$ respectively such that $n\ge p\ge 2$ or . Let $\mathcal{O}\left(N,P\right)$ denote an open subspace of $J^{\mathrm{\infty }}\left(N,P\right)$ which consists of all regular jets and singular jets of certain given $\mathcal{K}$-invariant class (including fold jets if $n\ge p$). An $\mathcal{O}$-regular map $f:N\to P$ refers to a smooth map such that $j^{\mathrm{\infty }}f\left(N\right)\subset \mathcal{O}\left(N,P\right)$. We will prove that a continuous section $s$ of $\mathcal{O}\left(N,P\right)$ over $N$ has an $\mathcal{O}\left(N,P\right)$-regular map $f$ such that $s$ and $j^{\mathrm{\infty }}f$ are homotopic as sections. As an application we will prove this homotopy principle for maps with $\mathcal{K}$-simple singularities of given class.

In this paper we investigate the relationship among the following integrals $${\int }_B\left|u\right(x\left){|}^{p-i}\right|\nabla u\left(x\right){|}^i(1-|x\left|{)}^{\alpha }dV\right(x),$$where $i\in \{0,1,2\}$, , $\alpha >0$, and where $u$ is an arbitrary harmonic function on the unit ball $B\subset {\mathbf{R}}^n$. Growth of the integral means of harmonic functions is also compared to the integral means of their gradient.