We study non-commutativity of the self homotopy groups of Lie groups.

A continued fraction C(-q,q) is defined as a special case of a general continued fraction F(a,b,c,{$\lambda$}q), which we have considered earlier in a separate paper. This continued fraction is also a special case of Ramanujan's continued fraction. In this paper we have found some very interesting q-identities and some identities analogous to identities given by Ramanujan involving G(-q,q) and H(-q,q) and one identity which gives the square of a continued fraction.

In terms of the exponential growth of a non-compact Riemannian manifold, we give an upper bounds for the bottom of the essential spectrum of the Laplacian. This is an improvement of Brooks' result.

We consider holomorphic families of rational maps from the viewpoint of complex dynamics.

First, we consider some classes of families of rational maps which satisfy a certain stability condition. We show a finiteness theorem for such holomorphic families of rational maps parameterized by a Riemann surface of finite type.

Next, we consider the monodromy of quasiconformally stable holomorphic families of rational maps over a punctured disk, and study the action of the monodromy on the Julia set.

In this paper, we prove that if a disjoint union of a countable number of complex affine subspaces is interpolating for the Hörmander algebra, then it can be written as the common zero set of {$\alpha$} + 1 functions in the Hörmander algebra, where {$\alpha$} is the maximum number of codimensions of the complex affine subspaces. Finally, we prove with an example in one complex variable that the number {$\alpha$} + 1 is lowest.

In this paper we study geodesics on curvature-adapted real hypersurfaces in non-flat quaternionic space forms. By observing the extrinsic shape of geodesics on these hypersurfaces we characterize them in the class of real hypersurfaces. We also investigate the length spectrum of geodesic spheres, which are the simplest curvature-adapted real hypersurfaces, in non-flat quaternionic space forms.

We give an explicit formula for the maximal polar quotient of a plane curve singularity with some applications to the Łojasiewicz exponent and the {$C\sp 0$}-degree of sufficiency.

In this paper, we prove the existence of orientable and nonorientable complete minimal surfaces of {${\bf R}\sp 3$} lying in a solid cylinder, a ball or a halfspace, using the Runge's approximation theorem and the Enneper-Weierstrass representation of minimal surfaces.