The aim of this paper is to investigate the phenomena of microlocal smoothing effect for Schrödinger type equations, in Gevrey spaces. We shall prove that microlocal Gevrey regularity of the solutions of the Cauchy problem for Schrödinger equation, depends on the initial data decay along a backward bicharacteristic.

In the local dynamics of Newton's method of a holomorphic function of two variables, a multiple root of rank 1 has a Cantor family of holomorphic superstable manifolds which consists of quadratically convergent initial values.

We prove that every Stieltjes problem has a solution in Gel'fand-Shilov spaces ${\mathcal{S}}^{\beta }$ for every $\beta >1$. In other words, for an arbitrary sequence $\left\{{\mu }_n\right\}$ there exists a function $\varphi$ in the Gel'fand-Shilov space ${\mathcal{S}}^{\beta }$ with support in the positive real line whose moment ${\int }_0^{\infty }{x}^n\varphi \left(x\right)dx={\mu }_n$ for every nonnegative integer $n$.

This improves the result of A. J. Duran in 1989 very much who showed that every Stieltjes moment problem has a solution in the Schwartz space $\mathcal{S}$, since the Gel'fand-Shilov space is much a smaller subspace of the Schwartz space. Duran's result already improved the result of R. P. Boas in 1939 who showed that every Stieltjes moment problem has a solution in the class of functions of bounded variation. Our result is optimal in a sense that if $\beta \le 1$ we cannot find a solution of the Stieltjes problem for a given sequence.

We use the half-space model for the open set of a de Sitter space associated to the steady state space to obtain some sharp a priori estimates for the height and the slope of certain constant mean curvature spacelike graphs. These estimates allow us to prove some existence and uniqueness theorems about complete non-compact constant mean curvature spacelike hypersurfaces in de Sitter spaces with prescribed asymptotic future boundary. Their geometric properties are studied.

A pass-move and a $♯$-move are local moves on oriented links defined by L. H. Kauffman and H. Murakami respectively. Two links are self pass-equivalent (resp. self $♯$-equivalent) if one can be deformed into the other by pass-moves (resp. #-moves), where none of them can occur between distinct components of the link. These relations are equivalence relations on ordered oriented links and stronger than link-homotopy defined by J. Milnor. We give two complete classifications of links with arbitrarily many components up to self pass-equivalence and up to self $♯$-equivalence respectively. So our classifications give subdivisions of link-homotopy classes.

We study the structure of the group of Lipschitz homeomorphisms of ${\mathbf{R}}^n$ leaving the origin fixed and the group of equivariant Lipschitz homeomorphisms of ${\mathbf{R}}^n$, and show that they are perfect. Next we apply these results for the groups of Lipschitz homeomorphisms of orbifolds and the groups of foliation preserving Lipschitz homeomorphisms for compact Hausdorff $C^1$-foliations.

There are many ways to give a rigorous meaning to the Feynman path integral. In the present paper especially the method of the time-slicing approximation determined through broken line paths is studied. It was proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in $L^2$ space as the discretization parameter tends to zero. In the present paper it is shown that these time-slicing approximate integrals converge in some weighted Sobolev spaces as well. Next as an application of this convergence result in the weighted Sobolev spaces, the path integral representation of correlation functions is studied of the position and the momentum operators. We note that their path integral representation is given in phase space. It is shown that the approximate integrals of correlation functions converge or diverge as the discretization parameter tends to zero. We note that the divergence of the approximate integrals reflects the uncertainty principle in quantum mechanics.

We give a formula for the coefficients of the Yablonskii-Vorob'ev polynomial. Also the reduction modulo a prime number of the polynomial is studied.

In this paper we will consider Fourier-Jacobi expansions of nearly holomorphic modular forms on $SU(2,1)$, whose coefficients are nearly holomorphic theta functions, and will construct a certain Galois action on them.

It is known that any compact connected Lie group with its left invariant framing is framed null-cobordant in the $p$-component for any prime $p\ne 2,3$. In this paper we will prove that the 3-components of $\mathrm{SO}\left(2n+1\right)$ and $Sp\left(n\right)$ are zero for $n\ge 3$, $n\ne 5,7,11$. Combining this with the previously known results on $\mathrm{SO}\left(2n\right)$ and $SU\left(n\right)$ consequently we see that any classical group has at most only the 2-component with some exceptions.

The purpose of this paper is to study extendibility and stable extendibility of vector bundles over real projective spaces. We determine a necessary and sufficient condition that a vector bundle $\zeta$ over the real projective $n$-space $R{P}^n$ is extendible (or stably extendible) to $R{P}^m$ for every $m>n$ in the case where $\zeta$ is the complexification of the tangent bundle of $R{P}^n$ and in the case where $\zeta$ is the normal bundle associated to an immersion of $R{P}^n$ in the Euclidean $(n+k)$-space $R^{n+k}$ or its complexification, and give examples of the normal bundle which is extendible to $R{P}^N$ but is not stably extendible to $R{P}^{N+1}$.

Let $G$ be a connected reductive algebraic group defined over a global field $k$ and $Q$ a maximal $k$-parabolic subgroup of $G$. The constant $\gamma (G,Q,k)$ attached to $(G,Q)$ is defined as an analogue of Hermite's constant. This constant depends only on $G,$$Q$ and $k$ in contrast to the previous definition of generalized Hermite constants ([W1]). Some functorial properties of $\gamma (G,Q,k)$ are proved. In the case that $k$ is a function field of one variable over a finite field, $\gamma (G{L}_n,Q,k)$ is computed.

We define the cobordism group of Morse functions on manifolds and show that it is an infinite cyclic group for dimension two in the oriented case. We also give an explicit Morse function which gives a generator of the group.

Let us consider the following nonlinear singular partial differential equation in the complex domain. When the equation is of totally characteristic type, the author has proved with H. Chen in [2] the existence of the unique holomorphic solution provided that the equation satisfies the Poincaré condition and that no resonances occur. In this paper, he will solve the same equation in the case where some resonances occur.

Let $T(·)$ be a (not necessarily exponentially bounded, not necessarily nondegenerate)$\alpha$-times integrated $C$-semigroup and let $-B$ be the generator of a $\left(C_0\right)\text{-}$ group $S(·)$ commuting with $T(·)$ and $C$. Under suitable conditions on $T(·)$ and $S(·)$ we prove the existence of an $\alpha$-times integrated $C$-semigroup $V(·)$, which has generator $\overline{A+B}$ provided that $T(·)$ is nondegenerate and has generator $A$. Explicit expressions of $V(·)$ in terms of $T(·)$ and $S(·)$ are obtained. In particular, when $B$ is bounded, $V(·)$ can be constructed by means of a series in terms of $T(·)$ and powers of $B$.