We study the analytic smoothing effect for a class of dispersive equations. In this paper we consider the microlocal analytic smoothness for the solutions of a class of dispersive equations including not only the Schrödinger equation but also the linearized $KdV$ equation. We make use of the Sjöstrand theory of the FBI transform as in Robbiano-Zuily's works in the case of Schrödinger equations.

In this paper, we deal with normal two-dimensional singularities with multiplicity two. We call such a singularity double point. The purpose of this paper is to give an estimate of the arithmetic genus for double points in terms of Horikawa's canonical resolution and a $p_{a}$-formula for some class of double points. It is known that, by using the data obtained from the canonical resolution, the geometric genus for double points is formulated, and rational double points and elliptic double points are characterized. We give a characterization of double points with the arithmetic genus two.

In this paper spaces with a $\sigma$-compact-finite weak base are discussed, and some characterizations of $g$-metrizable spaces are obtained by spaces with $\sigma$-compact-finite weak base and spaces with a $\sigma$-weakly hereditarily closure-preserved weak base.

It is well-known that every parallel isometric immersion of space forms into space forms is isotropic in the sense of O'Neill. We characterize this parallel immersion from this point of view.

Let $(M, \Omega)$ be a locally conformal symplectic manifold. $\Omega$ is a non-degenerate 2-form on $M$ such that there is a closed 1-form $\omega$, called the Lee form, satisfing $ d\Omega=\omega\wedge\Omega$. In this paper we consider Marsden-Weinstein reduction theorem which induces Jacobi-Liouville theorem as a special case. For locally conformal Kähler manifolds, this reduction theorem gives a construction of non-Kähler manifolds in general dimension.

In this paper we extend a well-known result concerning hypoellipticity and local solvability of linear partial differential operators on Schwartz distributions (see [14] and [19]) to the framework of pseudolocal continuous linear maps $T$ acting on Gevrey classes. Namely we prove that the Gevrey hypoellipticity of $T$ implies the Gevrey local solvability of the transposed operator $'T$. As an application, we identify some classes of non-Gevreyhypoelliptic operators. A fundamental kemel is also constructed for any Gevrey hypoelliptic partial differential operator.

Let $f$ : $X\rightarrow Y$ be a perfect surjective map of paracompact spaces. It is shown that if $Y$ is a C-space (resp., $\dim Y\leq n$ and $\dim f\leq m)$ , then the function space $C(X, I^{\infty})$ (resp., $C(X,$ $I^{2n+1+m})$) equipped with the source limitation topology contains a dense $G_{\delta}$-set $\mathscr{H}$ such that $f\times g$ embeds $X$ into $Y\times I^{\infty}$ (resp., into $Y\times I^{2n+1+m}$) for every $g\in \mathscr{H}$. Some applications of this result are also given.

An almost para-cosymplectic manifold is by definition an odd-dimensional differentiable manifold endowed with an almost paracontact stmcture with hyperbolic metric for which the structure forms are closed. The local structure of an almost para-cosymplectic manifold is described. We also treat some special subclasses of this class of manifolds: para-cosymplectic, weakly para-cosymplectic and almost para-cosymplectic with para-Kählerian leaves. Necessary and sufficient conditions for an almost para-cosymplectic manifold to be para-cosymplectic are found. Necessary and sufficient conditions for an almost para-cosymplectic manifold with para-Kählerian leaves to be weakly para-cosymplectic are also established. We construct examples of weakly para-cosymplectic manifolds, which are not paracosymplectic. It is proved that in dimensions $\geq 5$ an almost paracosymplectic manifold cannot be of non-zero constant sectional curvature. Main curvature identities which are fulfilled by any almost para-cosymplectic manifold are found.

The aim of this note is to construct explicitly a class of bi-Frobenius algebras via quivers. In particular, this kind of bi-Frobenius algebras are not Hopf algebras, and a necessary and sufficient condition for such algebras being symmetric is given.

A surface $S$ in $R^{3}$ is called parallel curved if there exists a plane such that at each point of $S$, there exists a principal direction parallel to this plane. In [2], we studied real-analytic, parallel curved surfaces and in particular, we showed that a connected, complete, real-analytic, embedded, parallel curved surface is homeomorphic to a sphere, a plane, a cylinder, or a torus. In the present paper, we shall show that a connected, complete, embedded, parallel curved surface such that any umbilical point is isolated is also homeomorphic to a sphere, a plane, a cylinder or a torus. However, we shall also show that for each non-negative integer $g\in N\cup\{0\}$, there exists a connected, compact, orientable, embedded, parallel curved surface of genus $g$.