We classify locally compact, geodesically complete, 2-dimensional Hadamard spaces whose Tits ideal boundaries have the minimal diameter $\pi$. Furthermore, we classify the universal covering spaces of certain 2-dimensional nonpositively curved spaces, which is an extension of the result obtained in the polyhedral case by W. Ballmann, M. Brin, and S. Barré.

We study expansive invertible onesided cellular automata (i.e., expansive automorphisms of onesided full shifts) and find severe dynamical and arithmetic constraints which provide partial answers to questions raised by M. Nasu [N2]. We employ the images and bilateral dimension groups, measure multipliers, and constructive combinatorial characterizations for two classes of cellular automata.

In this article the nonlinear equation of motion of vibrating membrane $u_{tt}-div\left\{{\sqrt{1+|Vu{|}^2}}^{-1}Vu\right\}=0$ is discussed in the space of functions having bounded variation. Approximate solutions are constructed in Rothe's method. It is proved that a subsequence of them converges to a function $u$ and that, if $u$ satisfies the energy conservation law, then it is a weak solution in the space of functions having bounded variation. The main tool is varifold convergence.

Consider a linear partial differential equation in $C^{d+1}P(z,\partial )u\left(z\right)=f\left(z\right)$, where $u\left(z\right)$ and $f\left(z\right)$ admit singularities on the surface $\{z_0=0\}$. We assume that $\left|f\right(z\left)\right|\le A|z_0{|}^c$ in some sectorial region with respect to $z_0$. We can give an exponent ${\gamma }^{*}>0$ for each operator $P(z,\partial )$ and show for those satisfying some conditions that if $\forall ϵ>0\exists C_{ϵ}$ such that $\left|u\right(z\left)\right|\le C_{ϵ}\exp \left(ϵ\right|z_0{|}^{-{\gamma }^{*}})$ in the sectorial region, then $\left|u\right(z\left)\right|\le C|z_0{|}^{c^{\prime }}$ for some constants $c^{\prime }$ and $C$.

We prove that logarithmic geometric genus of a complement of plane curve with logarithmic Kodaira dimension zero is equal to one.

We introduce a notion of double group construction within the category of quasi Woronowicz algebras which are regarded as quantum groups in the von Neumann algebra framework. We show that the quantum double in this setting is always unimodular. The Kac-Takesaki operator of the double group is explicitly described. It is also proven that the dual of the quantum double has a quasitriangular structure.

In this paper, considering the problem when the completion of a metric ANR $X$ is an ANR and $X$ is homotopy dense in the completion, we give some sufficient conditions. It is also shown that each uniform ANR is homotopy dense in any metric space containing $X$ isometrically as a dense subset, and that a metric space $X$ is a uniform ANR if and only if the metric completion of $X$ is a uniform ANR with $X$ a homotopy dense subset. Furthermore, introducing the notions of densely (local) hyper-connectedness and uniformly (local) hyper-connectedness, we characterize of AR's (ANR's) and uniform AR's (uniform ANR's), respectively.

We show that a Lie algebroid on a stratified manifold is integrable if, and only if, its restriction to each strata is integrable. These results allow us to construct a large class of algebras of pseudodifferential operators. They are also relevant for the definition of the graph of certain singular foliations of manifolds with corners and the construction of natural algebras of pseudodifferential operators on a given complex algebraic variety.

We investigate the cohomology of a group having finite virtual cohomological dimension in terms of the contributions from finite subgroups. As a result, we prove a variant of Quillen's $F$-isomorphism theorem which remains valid for an arbitrary commutative ring of coefficients and for suitable families of finite subgroups.

Our primary object of this paper is to give a representation formula for surfaces with prescribed mean curvature in the hyperbolic 3-space of curvature -1 in terms of their normal Gauss maps. For CMC (constant mean curvature) surfaces, we derive another representation formula in terms of their adjusted Gauss maps. These formulas are hyperbolic versions of the Kenmotsu representation formula for surfaces in the Euclidean 3-space. As an application, we give a construction of complete simply connected CMC surfaces embedded in the hyperbolic 3-space.

Some regularity properties of the Azukawa pseudometric for a wide class of domains (including bounded hyperconvex domains) are proven. Among others we prove that in this class of domains the Azukawa metric is continuous and the upper limit in its definition may be replaced with the limit. Some properties of the pluricomplex Green function with one (as well as with many) pole are also given.

Let $F_x$ denote the Farey series of order $\left[x\right]$, i.e. the increasing sequence of irreducible fractions $p_v\in (0,1]$ whose denominators do not exceed $x$. We shall obtain precise asymptotic formulae for the sum ${{\displaystyle \sum }}_{v=1}^{\Phi \left(x\right)}{p}_v^Z$ for complex $z$ and related sums, $\Phi \left(x\right)=♯F_x$ coinciding the summatory function of Euler's function. In particular, we shall prove an asymptotic formula for ${\displaystyle \sum }{p}_v^{-1}$ with as good an estimate as for the prime number theorem by extracting an intermediate error term occurring in the asymptotic formula for $\Phi \left(x\right)$.

In this paper, we classify the homotopy types of spaces of monic polynomials which have no $n$-fold real roots or spaces of $n$-tuples of monic polynomials which have no common real roots, by using the "scanning method"([9]) and Vassiliev's spectral sequence ([15], [16]). In particular, we show that such spaces are finite dimensional models for the infinite dimensional loop space of spheres.