Let $G$ be a finite abelian group and let $SK_{*}^{G}(pt,pt)$ be a cutting and pasting group (an SK group) based on $G$ manifolds with boundary. In this paper, we first obtain a basis for a $\mathbf{Z}$ module $\mathcal{T}_{*}^{G}$ consisting of all homomorphisms ($G$-SK invariants) $T:SK_{*}^{G}(pt,pt)\rightarrow\mathbf{Z}$. Let $SK_{*}^{G}$ be the SK group based on closed $G$ manifolds. We next study a relation between the theories $SK_{*}^{G}$ and $SK_{*}^{G}(pt,pt)$ by performing equivariant cuttings and pastings of $G$ manifolds, and characterize a class of multiplicative invariants which are related to $\chi^G$.

In [GW], S. B. Giddings and S. A. Wolpert proposed a procedure to obtain a new cell decomposition of the moduli space of curves. In this paper, we work out this procedure in detail. The number of cells in this new cell decomposition is smaller than that in other cell decompositions given in [BE, Ha, P3] and this makes the explicit computations of the orbifold Euler numbers of the moduli spaces for small genera easier. We check in many examples that they coincide with the known value.

We shall introduce four Hilbert spaces with a reproducing kernel and study relations among them. These Hilbert spaces are related to Fourier transforms of analytic functionals on the complex sphere.

An infinitary version of edge path groups is introduced for applications to non-locally simply connected spaces (see Figure 1 in the text). (1) Edge path groups in this paper are subgroups of the free $\sigma$-product of copies of the integer group $\mathbf{Z}$, which is isomorphic to the fundamental groups of the Hawaiian earring of $I$-many circles for some index set $I$. (2) Let $Y$ be a subspace of the real line in the Euclidean plane $\mathbf{R}^2$ and $\mathcal{C}$ the set of all connected components of $Y$. Then, the fundamental group of $\mathbf{R}^2\backslash Y$ is isomorphic to a free product of infinitely many non-trivial groups, if and only if there exists an accumulation point of $\mathcal{C}$ in $Y\cup\{\infty\}\cup-\infty$.

We prove the existence and uniqueness of mild and classical solutions of a quasilinear integrodifferential equation with nonlocal condition. The results are obtained by using $C_0$-semigroup and the Banach fixed point theorem.

We define the binormal indicatrix and focal developable of a nonlightlike curve in Minkowski 3-space. We establish the relationships between singularities of these subjects and geometric invariants of curves under the action of Lorentzian group.

The degenerate Neumann problem \[ \begin{cases} \ \displaystyle \sum_{i,j=1}^{n}a^{ij}(x)\frac{\partial^{2}u}{\partial x_i\partial x_j}=f(x,u,Du) & \text{in}\ \Omega ,\\ \ a(x)\dfrac{\partial u}{\partial v}+b(x)u=\varphi(x) & \text{on}\ \Gamma \end{cases} \] is studied in the case where $a(x)$ and $b(x)$ are non-negative functions on $\Gamma$ such that $a(x)+b(x)>0$ on $\Gamma$. A classical existence and uniqueness theorem in the Hölder space $C^{2+\alpha}(\bar{\Omega})$ is proved under suitable regularity and structure conditions on the data.

The class $L_\infty(b, Q)$ of completely operator semi-selfdecomposable distributions on $\mathbf{R}^d$ for $b$ and $Q$ is studied. Here $0<b<1$ and $Q$ is a $d\times d$ matrix whose eigenvalues have positive real parts. This is the limiting class of the decreasing sequence of classes $L_m(b, Q)$, $m=-1,0,1,\ldots$, where $L_{-1}(b, Q)$ is the class of all infinitely divisible distributions on $\mathbf{R}^d$ and $L_m(b,Q)$ is defined inductively as the class of distributions $\mu$ with characteristic function $\hat{\mu}(z)$ satisfying $\hat{\mu}(z)=\hat{\mu}(b^{Q'}z)\hat{\rho}(z)$ for some $\rho\in L_{m-1}(b,Q)$. $Q'$ is the transpose of $Q$. Distributions in $L_\infty(b,Q)$ are characterized in terms of Gaussian covariance matrices and Lévy measures. The connection with the class $OSS(b,Q)$ of operator semi-stable distributions on $\mathbf{R}^{d}$ for $b$ and $Q$ is established.