We obtain the first and second variational formulas for the volume function over the space of all equiaffine immersions. By using the first variational formula, we define some kinds of volume functions over the space of all non-degenerate immersions and obtain the Euler-Lagrange equation for those volume functions. Also, in the case where the ambient space is flat, we obtain the second variational formulas for special ones of those volume functions.

We obtain new lower bounds on the codimension of local isometric imbeddings of complex and quaternion projective spaces. We show that any open set of the complex projective space $P^n(\pmb{C})$ (resp. quaternion projective space $P^n(\pmb{H})$) cannot be locally isometrically imbedded into the euclidean space of dimension $4n-3$ (resp. $8n-4$). These estimates improve the previously known results obtained in [2] and [7].

In this note, we consider a maximal operator $\sup_{t \in \mathbb{R}}|u(x,t)| = \sup_{t \in \mathbb{R}}|e^{it\Omega(D)}f(x)|$, where $u$ is the solution to the initial value problem $u_t = i\Omega(D)u$, $u(0) = f$ for a $C^2$ function $\Omega$ with some growth rate at infinity. We prove that the operator $\sup_{t \in \mathbb{R}}|u(x,t)|$ has a mapping property from a fractional Sobolev space $H^\fraca{1}{4}$ with additional angular regularity in which the data lives to $L^2((1 + |x|)^{-b}dx) (b > 1)$ . This mapping property implies the almost everywhere convergence of $u(x,t)$ to $f$ as $t \to 0$, if the data $f$ has an angular regularity as well as $H^\frac{1}{4}$ regularity.

We prove that the 3 dimensional wave-Schr\"odinger system is globally well-posed for data in $(H^{s_{1}}\times \dot{H}^{s_{2}} \times \dot{H}^{s_{2}-1})(\R^{3})$, where both $s_{1}$ and $s_{2}$ are some negative indices.

We discuss local normal forms of Pfaffian systems and obtain a necessary and sufficient condition, in terms of relative polarizations, for the local generators of a Pfaffian system to convert to the contact system on the jet manifold $J^r(\R^h,\R^q)$, or to the Pfaffian system associated to a system of partial differential equations. This result generalizes the Darboux theorem on a Pfaffian equation of constant class.

In this paper we study asymptotic behavior of trajectories for a K\"ahler magnetic fields on a product of complex hyperbolic spaces and investigate the rank of a Hermitian symmetric space of noncompact type in terms of horocycle trajectories for K\"ahler magnetic fields.

We study the theory of scattering for the Zakharov system in space dimension 3. We prove in particular the existence of wave operators for that system with no size restriction on the data in larger spaces and for more general asymptotic states than were previously considered, and we determine convergence rates in time of solutions in the range of the wave operators to the solutions of the underlying linear system. We also consider the same system in space dimension 2, where we prove the existence of wave operators for small Schr\"odinger data in the special case of vanishing asymptotic data for the wave field.

The main purpose of this paper is to determine the fine spectrum of the operator $B(r,s)$ defined by a band matrix over the sequence spaces $\ell_1$ and $bv$. Since the operator $B(r,s)$ is reduced to the right-shift, difference and Zweier matrices in special cases $(r,s)=(0,1),(r,s)=(1,-1) $ and $(r,s)=(r,1-r)$, respectively, our results are more general than the corresponding results in the existing literature.

In the previous paper [20], we have classified generic holonomic systems of general Clairaut type. The notion of holonomic systems of general Clairaut type is one of the generalized notions of the classical Clairaut equations and Clairaut type. We give a generic classification of bifurcations of them as an application of the theory of complete Legendrian unfoldings and transversality theorem. In the list of normal forms, the Lagrangian equation appears. Moreover, there appears several new equations.

Let $D$ be a proper subdomain of Euclidean $n$-space $R^n\ (n\geq2)$. The following two necessary and sufficient conditions for uniform domains are obtained in this paper: (1) $D$ is a uniform domain if and only if there exists a constant $m=m(D)$ such that $k_D(x_1,x_2)\leq mj_D(x_1,x_2)$ for any $x_1, x_2\in D$, where $k_D$ is the quasi-hyperbolic metric in $D$, $j_D(x_1,x_2)=\frac12\log\left(\frac{|x_1-x_2|}{d(x_1,\partial D)}+1\right)\left(\frac{|x_1-x_2|}{d(x_2,\partial D)}+1\right)$. (2) $D$ is a uniform domain if and only if there exists a constant $M=M(D)$ such that each pair of points $x_1,x_2\in D$ can be joined by a rectifiable arc $\gamma\subset D$ which satisfies $\frac1{\left(c_2^\alpha-c_1^\alpha\right)}\int_{\gamma_{j,[c_1,c_2]}}d(x,\partial D)^{\alpha-1}\mbox{d}s\leq\frac M\alpha|x_1-x_2|^\alpha$ for any $0<\alpha\leq1$ and $0\leq c_1< c_2\leq1/2$, $j=1,2$, where $\gamma_{j,[c_1,c_2]}$ denotes the subarc between $\gamma_j(c_1l(\gamma))$ and $\gamma_j(c_2l(\gamma))$, $\gamma_j$ is the arc $\gamma$ which starts from $x_j$ and use arc length $s$ as parameter, $l(\gamma)$ is the Euclidean length of $\gamma$.