A classification is given for edge transitive circulant graphs whose complements are also edge transitive. In particular, it is shown that the only self-complementary symmetric circulant graphs are certain Paley graphs with a prime number of vertices. Similar results on digraphs are also obtained.

We shall consider compact complex manifolds of dimension three which have subdomains in a three dimensional projective space as their unramified even coverings. Assume that the subdomains contain projective lines. Then any pair of such manifolds can be connected together complex analytically by an analogue of Klein combinations [K1]. In this paper, we shall prove a (weak) analogue of two results of B. Maskit [M] on the Lebesgue measures of the limit set of Kleinian groups (Theorems A and B). In section 1, we shall give definitions of terms and precise statement of our result. In section 2, we shall analyze the limit set by the same method as that of Maskit. Section 3 is devoted to proving Theorem 2.1. As a corollary to Theorem 2.1, we obtain Theorem A, which is an analogue of Combination Theorem I of [M]. In section 4, we shall prove Theorem B, which is an analogue of Combination Theorem II of [M].

A mixed problem for some hyperbolic equation with small parameter $\varepsilon$ under the presence of a restoring term $|u|^{\alpha}u$ and a reduced problem for a parabolic type are considered. Several $\varepsilon$ weighted energy estimates can be obtained by the method of difference quotients. It is shown that the solution $u_\varepsilon$ of the mixed problem converges, uniformly on any finite time interval, to the solution $u$ of the problem for the parabolic equation in an appropriate Hilbert space as $\varepsilon\to 0$.

We show that the algebraic structure of the group $C^*$-algebra $C^*(G)$ of a simply connected, connected nilpotent Lie group $G$ is described as repeating finitely the extension of $C^*$-algebras with $T_{2^-}$ spectrums by themselves and one more extension by a commutative $C^*$-algebra on the fixed point space $(\mathfrak{G}^*)^G$ of $\mathfrak{G}^*$ under the coadjoint action of $G$. Using this result, we show that $C^*(G)$ has no non-trivial projections.

We consider surfaces in simply connected 4-manifolds. We estimate the normal Euler numbers of bounded non-orientable surfaces and consider the problem of representing characteristic homology classes by orientable surfaces. To do so, we develop techniques connecting the above problems for given surfaces with the problems for surfaces with fewer first Betti numbers.