We show a correspondence between the set of all $G$-invariant projectively flat connections on a homogeneous space $M=G/K$, and the one of all $\widetilde{G}$-invariant flat connections on homogeneous spaces $\widetilde{M}=\widetilde{G}/K$, where $\widetilde{G}$ is a central extension of $G$ (Theorem 3.3).

The Fatou theorem on the Poisson representation of bounded harmonic functions on a half space is generalized to the bounded solutions $u(t)$ of the second order equation $$ u''(t) = A u(t), 0 < t < \infty, $$ in a dual Banach space $X = X_*{'}$, when $A$ is the dual of a non-negative operator $A_*$ with dense domain in $X_*$. Any bounded weak* solution is represented as $u(t) =$ $\exp(-t\sqrt{A})f$ with the weak* initial value $f$. Its prototype is in A.~V. Balakrishnan's paper in 1960 on fractional powers of non-negative operators. This is applied to prove the uniqueness of solutions in the theory of signal transmission on submarine cables by W. Thomson in 1855.

We shall give some remarks on a theorem of Benard.

A solvable graph of a finite group is one of generalized prime graphs of groups which are introduced as generalizations of a prime graph of a finite group in \cite{AI1}. In this paper we will characterize a Fermat prime by a solvable graph of a finite group.

Let $U$ be a bounded symmetric open neighborhood of the origin of $\mathbf{R}^{m+k} \ (k\geqq 1)$. We shall prove a generalization of the Borsuk's antipodal theorem for an admissible mapping $\varphi:\partial\overline{U}\to \mathbf{R}^m$ and the related topic. We shall generalize the theorem for the case of a bounded symmetric open neighborhood $U$ of the origin of an infinite dimensional normed space $\mathbf{E}$. The Borsuk-Ulam theorem shall be studied for the case of a bounded symmetric open neighborhood $U$ of the origin of an infinite dimensional normed space $\mathbf{E}$.

In this paper we discuss the limiting absorption principle (l.a.p.) of the second quantization of semi-bounded self-adjoint operators. We show that the l.a.p. for a self-adjoint operator on a basic Hilbert space $\mathcal{H}$ is ``inherited'' to the one for its second quantization on a Fock space $\mathcal{F}(\mathcal{H})$. In order to show such a result, we examine the resolvent of $n$-body problem and take the limit of the infinite direct sum of those operators in a suitable subspace of $\mathcal{F}(\mathcal{H})$.