In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the δ-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel ([BG]). The second criterion is based on the I-equisingularity of a k-parametric family (k ≥ 1) of generically reduced plane curve singularities, which is introduced by Nobile ([No]) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studied.

Let H be the upper half-space of the Euclidean space. The α-parabolic Bloch type space $¥cal B$_α(σ) on H is the set of all solutions u of the parabolic equation (∂/∂t + (−Δ_x)^α)u = 0 with 0 < α ≤ 1 which belong to C¹(H) and have finite Bloch norm with weight t^σ. In this paper, we study representing and interpolating sequences on parabolic Bloch type spaces. In our previous paper [8], the reproducing formula on $¥cal B$_α(σ) is given. A representing sequence gives a discrete version of the reproducing formula on $¥cal B$_α(σ). Interpolating sequences are closely related to representing sequences, and such sequences are very interesting in their own right.

An elliptic fibration over P₃($¥mathbb{C}$), naively arising from the Euler equation for free rigid body dynamics, is studied from the viewpoint of complex algebraic geometry. With this elliptic fibration, associated is an elliptic fibration in Weierstraß normal form, whose generic fibres are isomorphic to those of the original fibration. This normal form is desingularized in a canonical manner. It is shown that there is a four-to-one meromorphic mapping from the naive elliptic fibration to the Weierstraß mormal form. The latter fibration is also shown to be bimeromorphic to the family of spectral curves arising from the corresponding Manakov equation.

It is well known that a Monge-Ampère equation can be expressed in terms of exterior differential system—Monge-Ampère system, which is the ideal generated algebraically by a contact form and a 2-form and its exterior derivatives on a 5-dimensional contact manifold, and the system is independent of the choice of coordinate system. On the other hand, a single second order partial differential equation of one unknown function with two independent variables corresponds to the differential system on a hypersurface of Lagrange-Grassmann bundle over a 5-dimensional contact manifold obtained by restricting its canonical system to the hypersurface. We observe relations between Monge characteristic systems of Monge-Ampère equation and those of Monge-Ampère system and particularly analyze structure equations of those systems. This observation leads to the result—to characterize Monge-Ampère equation by the property that the certain differential system defined from the Monge characteristic system drops down to the contact manifold.

In this paper, the definition of generalized spacelike Mannheim curve in Minkowski space-time E₁⁴ is given. The necessary and sufficient conditions for the generalized spacelike Mannheim curve are obtained. Also, some characterizations of Mannheim curve are given.