We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal $G$-bundle over a closed Lipschitz manifold is perfect when $G$ is a compact Lie group.

Let $N\ge 1$ and $p>1$. Let $F$ be a compact set and $\Omega$ be a bounded open set of $R^N$ satisfying $F\subset \Omega \subset R^N$. We define a generalized $p$-harmonic operator $L_p$ which is elliptic in $\Omega \setminus F$ and degenerated on $F$. We shall study the genuinely degenerate elliptic equations with absorption term. In connection with these equations we shall treat two topics in the present paper. Namely, the one is concerned with removable singularities of solutions and the other is the unique existence property of bounded solutions for the Dirichlet boundary problem.

Let $V$ be a simple vertex operator algebra and Aut $V$ a finite abelian subgroup such that $V^G$ is rational. We study the representations of $V$ based on certain assumptions on $V^G$-modules. We prove a decomposition theorem for irreducible $V$-modules. We also define an induced module from $V^G$ to $V$ and show that every irreducible $V$-module is a quotient module of some induced module. In addition, we prove that $V$ is rational in this case.

We describe 4-dimensional complex projective manifolds $X$ admitting a simple normal crossing divisor of the form $A+B$ among their hyperplane sections, both components $A$ and $B$ having sectional genus zero. Let $L$ be the hyperplane bundle. Up to exchanging the two components, $(X,L,A,B)$ is one of the following: 1)$(X,L)$ is a scroll over $P^1$ with $A$ itself a scroll and $B$ a fibre, 2)$(X,L)=(P^2\times P^2,{\mathcal{O}}_{P^2\times P^2}(1,1\left)\right)$ with $A\in \left|{\mathcal{O}}_{P^2\times P^2}\right(1,0\left)\right|,$$B\in \left|{\mathcal{O}}_{P^2\times P^2}\right(0,1\left)\right|,$$3)X=P_{P^2}(\mathcal{V})$ where $\mathcal{V}={\mathcal{O}}_{P^2}(1{)}^{\oplus 2}\oplus {\mathcal{O}}_{P^2}(2)$, $L$ is the tautological line bundle, $A=P_{P^2}$ $({\mathcal{O}}_{P^2}{(1)}^{\oplus 2}$, and $B\in {\pi }^{*}$$\left|{\mathcal{O}}_{P^2}\right(2\left)\right|$ , where $\pi$ : $X\to P^2$ is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.

It is known that the formal solution to an equation of non-Kowalevski type is divergent in general. To this divergent solution it is important to evaluate the rate of divergence or the Gevrey order, and such a result is often called a Maillet type theorem. In this paper the Maillet type theorem is proved for divergent solutions to singular partial differential equations of non-Kowalevski type, and it is shown that the Gevrey order is characterized by a Newton polygon associated with an equation. In order to prove our results the majorant method is effectively employed.

Singular invariant hyperfunctions on the space of $n\times n$ complex and quaternion matrices are discussed in this paper. Following a parallel method employed in the author's paper on invariant hyperfunctions on the symmetric matrix spaces, we give an algorithm to determine the orders of poles of the complex power of the determinant function and to determine exactly the support of singular invariant hyperfunctions, i.e., invariant hyperfunctions whose supports are contained in the set of points of rank strictly less than $n$, obtained as negative-order-coefficients of the Laurent expansions of the complex powers.

We investigate Lagrangian submanifolds of the 3-dimensional complex projective space. In case the second fundamental form takes a special form, we obtain several classification theorems. As a consequence we obtain several new examples of 3-dimensional Lagrangian submanifolds.

We give a necessary and sufficient condition for the existence of global solutions of some partial differential equation which is locally solvable and give some applications in complex analysis of several variables.

Magnetic potentials have a direct significance to the motion of particles in quantum mechanics. This is known as the Aharonov-Bohm effect. We study this quantum property in the scattering by two magnetic fields at large separation in two dimensions. We derive the asymptotic formula of scattering amplitude when the distance between the centers of two fields goes to infinity. The result depends on the fluxes of fields and on incident and final directions. We here consider only the simple case that at least one of two fluxes is zero.

We study the $n$-dimensional category $ca{t}_n\left(X\right)$ of a compact space $X$, a counterpart to Lusternik-Schnirelmann category in the context of $n$-homotopy theory, and prove Menger manifold analogues of results due to Montejano and Wong for Hilbert cube manifolds.

In this paper we study the Hankel transformation on Hardy type spaces. We also investigate Hankel convolution operators and Hankel multipliers on these Hardy spaces.

Let us consider the following nonlinear singular partial differential equation: in the complex domain. Denote by ${\mathcal{S}}_{+}$ [resp. ${\mathcal{S}}_{log}$] the set of all the solutions $u(t,x)$ with asymptotics $u(t,x)=O\left(\right|t{|}^a)$ [resp. $u(t,x)=O(1/|\log t{|}^a\left)\right]$(as $t\to O$ uniformly in $x$) for some $a>0$. Clearly ${\mathcal{S}}_{log}\supset {\mathcal{S}}_{+}$. The paper gives a sufficient condition for ${\mathcal{S}}_{log}={\mathcal{S}}_{+}$ to be valid.

Let $S$ be a smooth hypersurface in the projective three space and consider a projection of $S$ from $P\in S$ to a plane $H$. This projection induces an extension of fields $k\left(S\right)/k\left(H\right)$. The point $P$ is called a Galois point if the extension is Galois. We study structures of quartic surfaces focusing on Galois points. We will show that the number of the Galois points is zero, one, two, four or eight and the existence of some rule of distribution of the Galois points.