We prove a local (in time) unique vorticity existence for the Euler equation of an ideal incompressible fluid in a critical Besov space $\mathbf{B}_{\infty , 1}^{0}(\mathbb{R}^3) \cap \mathbf{L}^{p}(\mathbb{R}^3)$ with the initial vorticity $\omega _{0} \in \mathbf{B}_{\infty , 1}^{0}(\mathbb{R}^3) \cap \mathbf{L}^{p}(\mathbb{R}^3)$ for some $1 < p < 3$.

We give a criterion for spherical 3-manifolds to be obtained by surgery on knots in the 3-sphere in terms of an integral lift of the Rochlin invariant of them.

We shall deal with IBVP for a linear one-dimensional wave equation in domain with time-quasiperiodically oscillating boundaries. We shall show that for any given initial data and almost all boundary data, every solution is quasiperiodic in $t$, provided that the basic frequencies of timequasiperiodic data of IBVP satisfy the number-theoretic Diophantine conditions. In order to solve this problem, we shall show the reduction theorem of one-dimensional quasiperiodic dynamical systems. To prove the reduction theorem, we shall define upper and lower rotation numbers of dynamical systems and apply the rapidly iteration method to the related dynamical system defined by the boundary functions. Also we shall construct a class of time-quasiperiodic boundary data of IBVP and the basic frequencies such that IBVP has quasiperiodic solutions that are the superposition of the sequentially time-unbounded forward and backward waves.

Shintani expressed the first derivative at $s = 0$ of a partial $\zeta$-function of an algebraic number field in terms of the multiple gamma function. Cassou-Noguès constructed a $p$-adic analogue of the partial $\zeta$-function and calculated the derivative at $s = 0$. In this paper, we will define a $p$-adic analogue of the multiple gamma function and give a $p$-adic analogue of Shintani’s formula. This formula has a strong resemblance to the original Shintani’s formula. Using this formula, we get a partial result toward Gross’ conjecture concerning the order at $s = 0$ of the $p$-adic $L$-function.

The author proposed an idea on the principal part of linear systems of pseudo-differential equations in the Cauchy problem through the (pseudo-)normal form of systems in the formal symbol class and the theory of weighted determinant in [18]. In this paper, we show the regularity of the symbols of the principal part as $p$-evolution. In order to show this, we consider the regularity of the $p$-determinant of the matrices of pseudo-differential operators.

In this paper we characterize the degree 4 multiple lines with generic embedding dimension 3 and among them the ones with very degenerate hyperplane section, and the ones which contain a degree 3 planar subcurve. Using that characterization, we prove that the degree 4 curves containing a planar subcurve of degree 3 are the general element of an irreducible component of the Hilbert scheme. Moreover, we show that all the multiple lines we consider belong to the same connected component of the corresponding Hilbert scheme.

In their 1995 paper [3], Fehlmann and Gardiner posed an extremal problem, which will be called an obstacle problem throughout the present paper, for quadratic differentials on Riemann surfaces. A compact set $E$ in a Riemann surface $S$ of finite type is called an obstacle if each component of $E$ is relatively contractible in $S$ and if $S \backslash E$ is connected. For a given obstacle $E$ and a symmetric integrable quadratic differential $\varphi \neq 0$ on $S$; the obstacle problem is to find a conformal embedding $g$ of $S \backslash E$ into another Riemann surface $R$ of the same type as $S$ and a symmetric quadratic differential $\psi$ on $R$ so that the following three conditions hold: (i) the borders and punctures are preserved under the mapping $g$; (ii) the pull-back $g^{*}\psi$ gives the same heights vector as that of $\varphi$; and (iii) the norm $\| \psi \|_{L^{1}(R)}$ is maximal among those embeddings. Fehlmann and Gardiner asserted existence and uniqueness of a solution to the obstacle problem when $E$ consists only of finitely many components. It seems, however, that the uniqueness assertion is not correct in their form. In the present paper, we extend the existence theorem and give a correction to the uniqueness assertion in the general case. As an application we provide a slit mapping theorem for an open Riemann surface of finite genus.

In this paper, the global existence and asymptotic behavior in time of solutions for the nonlinear Schrödinger equation with the Stark effect in one or two space dimensions are studied. The nonlinearity is cubic and quadratic in one and two dimensional cases, respectively, and it is a summation of a gauge invariant term and non-gauge invariant terms. This nonlinearity is critical between the short range scattering and the long range one. A modified wave operator to this equation is constructed for small final states. Its domain is a certain small ball in $H^{2} \cap \mathcal{F}H^{2}$, where $\mathcal{F}$ is the Fourier transform.