The following inequalities between transfinite large inductive dimensions modulo absolutely additive (resp. multiplicative) Borel classes $A(\alpha)$ (resp. $M(\alpha))$ = \begin{equation*} (i) \; \; \; A(0) \mathsf{-trInd} \geq M(0) \mathsf{-trInd} \geq max\{A(1) \mathsf{-trInd}, M(1) \mathsf{-trInd}\}, and \\ (ii) \; \; \; min \{A(\alpha) \mathsf{-trInd},M(\alpha) \mathsf{-trInd}\} \geq max \{A(\beta) \mathsf{-trInd},M(\beta) \mathsf{-trInd}\}, \\ \mathsf{where \;} 1 \leq \alpha "< \beta < \omega_1. \end{equation*} We show that for any two functions $a$ and $m$ from the set of ordinals $\Omega = \{ \alpha : \alpha < \omega_1 \}$ to the set $\{ -1 \} \cup \Omega \cup \{ \infty \}$ such that

\begin{equation*} (i) \; \; \; a(0) \geq m(0) \geq max\{a(1),m(1)\}, \mathsf{and}\\ (ii) \; \; \; min \{a(\alpha),m(\alpha) \} \geq max \{ a(\beta),m(\beta)\}, \mathsf{whenever \;} 1 \geq \alpha < \beta < \omega_1, \end{equation*} there is a separable metrizable space X such that $A(\alpha) \mathsf{-trInd}X = a(/alpha)$ and $M(\alpha) \mathsf{-trInd}X = m(\alpha)$ for each $0 \geq \alpha < \omega_1$.

The extremal function $c_K$ for the variational 2-capacity $\mathrm{cap}(K)$ of a compact subset $K$ of the Royden harmonic boundary $\delta R$ of an open Riemann surface $R$ relative to an end $W$ of $R$, referred to as the capacitary function of $K$, is characterized as the Dirichlet finite harmonic function $h$ on $W$ vanishing continuously on the relative boundary $\partial W$ of $W$ satisfying the following three properties: the normal derivative measure $*\text{dh}$ of $h$ exists on $\delta R$ with $*\text{dh}\geqq 0$ on $\delta R$; $*\text{dh}=0$ on $\delta R\setminus K$; $h=1$ quasieverywhere on $K$. As a simple application of the above characterization, we will show the validity of the following inequality

$$\text{hm}\left(K\right)\leqq \kappa •\text{cap}{\left(K\right)}^{1/2}$$

for every compact subset $K$ of $\delta R$, where $\mathrm{hm}(K)$ is the harmonic measure of $K$ calculated at a fixed point $a$ in $W$ and $\kappa$ is a constant depending only upon the triple $(R,W,a)$.

Using a criterion of Johnson-Rees [9] we give a list of all four and six dimensional flat Kähler manifolds. We calculate their $\mathit{R}$–cohomology, including the Hodge numbers. As a corollary, we classify all flat complex manifolds of dimension 3 whose holonomy groups are subgroups of $\mathrm{SU}(3)$. Moreover, we define a family of flat Kähler manifolds which are generalizations of the oriented Hantzsche-Wendt Riemannian manifolds [14].

We show that any rosy CM-trivial theory has weak canonical bases, and CM-triviality in the real sort is equivalent to CM-triviality with geometric elimination of imaginaries. We also show that CM-triviality is equivalent to the modularity in O-minimal theories with elimination of imaginaries.

This paper classifies all $\mathit{Q}$-homology planes which appear as cyclic covers of ${\mathit{A}}^2$.

We consider the group of $C^{\infty }$-diffeomorphisms of $M$ which is isotopic to the identity through $C^{\infty }$-diffeomorphisms preserving $N$ for a compact manifold pair $(M,N)$ and prove that the group is perfect. Also we prove that it is uniformly perfect for a certain compact manifold with boundary.

An austere submanifold is a minimal submanifold where for each normal vector, the set of eigenvalues of its shape operator is invariant under the multiplication by $-1$. In the present paper, we introduce the notion of weakly reflective submanifold, which is an austere submanifold with a reflection for each normal direction, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of linear isotropy representations of Riemannian symmetric spaces.

In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps from mm-spaces to $\mathit{R}$-trees. Our main theorems assert that the concentration to $\mathit{R}$-trees is equivalent to the concentration to the real line.

In the homotopy theory of a 2-category with zeros and having a suspension functor we establish various composition properties of box brackets, including new formulae involving 2-sided matrix Toda brackets and classical Toda brackets. We are lead to define and study a new secondary homotopy operation called the box quartet operation. In the topological category this operation satisfies two triviality properties, one of which may be viewed as the foundation upon which an important classical mod zero result on Toda brackets rests. New insights and computations in the homotopy groups of spheres are obtained.

The classical Hardy theorem on $\mathit{R}$, which asserts $f$ and the Fourier transform of $f$ cannot both be very small, was generalized by Miyachi in terms of $L^1+L^{\infty }$ and ${log}^{+}$-functions. In this paper we generalize Miyachi's theorem for ${\mathit{R}}^d$ and then for other generalized Fourier transforms such as the Chébli-Trimèche and the Dunkl transforms.

The hyperbolic Schwarz map is defined in [SYY1] as a map from the complex projective line to the three-dimensional real hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front ([GMM], [KUY], [KRSUY]), and generic singularities are cuspidal edges and swallowtail singularities. In this paper, for the two-parameter family of the confluent hypergeometric differential equations, we study the singularities of the hyperbolic Schwarz map, count the number of swallowtails, and identify the further singularities, except those which are apparently of type $A_5$. This describes creations/eliminations of the swallowtails on the image surfaces, and gives a stratification of the parameter space according to types of singularities. Such a study was made for a 1-parameter family of hypergeometric differential equation in [NSYY], which counts only the number of swallowtails without identifying further singularities.

A diagram of a 2-knot consists of a finite number of compact, connected surfaces called sheets. We prove that if a 2-knot admits a non-trivial coloring by some quandle, then any diagram of the 2-knot needs at least four sheets. Moreover, if a 2-knot admits a non-trivial 5- or 7-coloring, then any diagram needs at least five or six sheets, respectively.

A compact Klein surface $X=\mathit{D}/\Gamma$, where $\mathit{D}$ denotes the hyperbolic plane and $\Gamma$ is a surface NEC group, is said to be $q$-trigonal if it admits an automorphism $\varphi$ of order $3$ such that the quotient has algebraic genus $q$. In this paper we obtain for each $q$ the automorphism groups of $q$-trigonal planar Klein surfaces, that is surfaces of topological genus $0$ with $k\ge 3$ boundary components. We also study the surfaces in this family, which have an automorphism group of maximal order (maximal surfaces). It will be done from an algebraic and geometrical point of view.

We show that the nonlinear wave equation $\square u+u_t^3=0$ is globally well-posed in radially symmetric Sobolev spaces $H_{\mathrm{rad}}^k({\mathit{R}}^3)\times H_{\mathrm{rad}}^{k-1}(R^3)$ for all integers $k>2$. This partially extends the well-posedness in $H^k({\mathit{R}}^3)\times H^{k-1}(R^3)$ for all $k\in [1,2]$, established by Lions and Strauss[12]. As a consequence we obtain the global existence of $C^{\infty }$ solutions with radial $C_0^{\infty }$ data. The regularity problem requires smoothing and non-concentration estimates in addition to standard energy estimates, since the cubic damping is critical when $k=2$. We also establish scattering results for initial data $(u,u_t){|}_{t=0}$ in radially symmetric Sobolev spaces.