We study the nonlinear magnetic Schrödinger equation, $-(\nabla -i\lambda A)^{2}u=f(x,|u|^{2})u$ on $\mathbb{R}^{N}$, where $N \geq 2$ and the nonlinearity is super-linear and subcritical. The vector potential $A$ and the associated magnetic field are assumed to vanish on a common bounded open set $\Omega$. It is shown that the equation above has more and more solutions which are localized near $\Omega$ as $\lambda \to \infty$.

We study regularizations of Schwartz distributions on a complete Riemannian manifold $M$. These approximations are based on families of smoothing operators obtained from the solution operator to the wave equation on $M$ derived from the metric Laplacian. The resulting global regularization processes are optimal in the sense that they preserve the microlocal structure of distributions, commute with isometries and provide sheaf embeddings into algebras of generalized functions on $M$.

Let $G$ be a simple algebraic group over $\mathbb{C}$. By taking the quasi-classical limit of the ring of differential operators on the corresponding quantized algebraic group at roots of 1 we obtain a Poisson manifold $\Delta G\times K$, where $\Delta G$ is the subgroup of $G\times G$ consisting of the diagonal elements, and $K$ is a certain subgroup of $G\times G$. We show that this Poisson structure coincides with the one introduced by Semenov-Tyan-Shansky geometrically in the framework of Manin triples.

Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered knots.

We present a Robinson-Schensted-Knuth type one-to-one correspondence between the set of pictures and the set of pairs of Littlewood-Richardson crystals.

In the present paper, we consider sampling theorems on Banach function spaces. Here we obtain a necessary and sufficient condition. For the latter half of the paper, we consider sampling theorems in terms of wavelets.

In [3], local moves, called {\it simple ribbon moves} for links are defined. In this paper, we study primeness of knots which can be transformed into the trivial knot by a single simple ribbon move.

In this article, we study a class of groups which are commensurable with a direct product of the discrete Heisenberg group and a free abelian group, or a free abelian group by using zeta functions of groups defined by Grunewald, Segal, and Smith as generating functions of the number of subgroups of given index $n$. We will show that zeta functions determine their isomorphism classes for groups belonging to the above class.

This paper is concerned with invariant densities for transformations on $\mathbb{R}$ which are the boundary restrictions of inner functions of the upper half plane. G. Letac [9] proved that if the corresponding inner function has a fixed point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ or a periodic point $z_{0}$ in $\mathbb{C}\setminus \mathbb{R}$ with period 2, then a Cauchy distribution $(1/\pi)\mathrm{Im}\left(1/(x-z_{0}) \right)$ is an invariant probability density for the transformation. Using Cauchy's integral formula, we give an easier proof of Letac's result. An easy sufficient condition for such transformations to be isomorphic to piecewise expanding transformations on an finite interval is given by the explicit form of the density. Transformations of the forms $\alpha x + \beta - \sum ^{n }_{k=1}b_{k}/(x-a_{k})$, \:$\alpha x-\sum ^{\infty }_{k=1}\left\{ b_{k}/(x-a_{k})+b_{k}/(x+a_{k}) \right\}$ and $\alpha x +\beta\tan x$ are studied as examples.

Let $\mathfrak{a}$ be an ideal of a local ring $(R,\mathfrak{m})$ and $X$ a $d$-dimensional homologically bounded complex of $R$-modules whose all homology modules are finitely generated. We show that $H^d_{\mathfrak{a}}(X)=0$ if and only if $\dim \widehat{R}/\mathfrak{a} \widehat{R}+\mathfrak{p}>0$ for all prime ideals $\mathfrak{p}$ of $\hat{R}$ such that $\dim \hat{R}/\mathfrak{p}-\inf (X\otimes_R\hat{R})_{\mathfrak{p}}=d$.

A family of $S^{1}$-equivariant hypersurfaces of constant mean curvature can be obtained by using the Lagrangians with suitable potentials in the hyperbolic 3-space. The conservation law is effectively applied to the construction of $S^{1}$-equivariant hypersurfaces of constant mean curvature in the hyperbolic 3-space.

Ōuchi ([2], [3]) found a formal solution $\widetilde{u}(t,x)=\sum_{k\geq 0}u_k(x)t^k$ with \begin{equation*} |u_k(x)|\leq AB^k\Gamma\Bigg(\frac{k}{\gamma_*}+1\Bigg) \quad 0<\gamma_*\leq\infty \end{equation*} for some class of nonlinear partial differential equations. For these equations he showed that there exists a genuine solution $u_{S}(t,x)$ on a sector $S$ with asymptotic expansion $u_{S}(t,x)\sim \widetilde{u}(t,x)$ as $t\rightarrow 0$ in the sector $S$. These equations have polynomial type nonlinear terms. In this paper we study a similar class of equations with the following nonlinear terms \begin{equation*} \sum_{|q|\geq1}t^{\sigma_q}c_q(t,x)\prod_{j+|\alpha|\leq m} \Bigg\{\Bigg(t\frac{\partial}{\partial t}\Bigg)^j\Bigg(\frac{\partial}{\partial x}\Bigg)^{\alpha}u(t,x)\Bigg\}^{q_{j,\alpha}}\,. \end{equation*} It is main purpose to get a solvability of the equation in a category $u_S(t,x)\sim 0$ as $t\rightarrow 0$ in a sector $S$. We give a proof by the method that is a little different from that in [3]. Further we give a remark that the similar class of equations has a genuine solution $u_{S}(t,x)$ with $u_{S}(t,x)\sim \widetilde{u}(t,x)$ as $t\rightarrow 0$ in the sector $S$.

Let $\xi=\sqrt{v/u}\tanh(uv)^{-1/2}$, where $u$ and $v$ are positive integers, and let $\eta=|h(\xi)|$, where $h(t)$ is a non-constant rational function with algebraic coefficients. We compute upper and lower bounds for the approximation of certain values $\eta$ of hyperbolic functions by rationals $x/y$ such that $x$ and $y$ satisfy Diophantine equations. We show that there are infinitely many coprime integers $x$ and $y$ such that $|y\eta-x|\ll\log\log y/\log y$ and a Diophantine equation holds simultaneously relating $x$ and $y$ and some integer $z$. Conversely, all positive integers $x$ and $y$ with $y\ge c_0$ solving the Diophantine equation satisfy $|y\eta-x|\gg\log\log y/\log y$.

We consider three kinds of discrete Sobolev inequalities corresponding to a graph Laplacian $\boldsymbol{A}$ on regular $M$-hedron for $M=4,6,8,12,20$. Discrete heat kernel $\boldsymbol{H}(t)=\exp(-t\boldsymbol{A})$, Green matrix $\boldsymbol{G}(a)=(\boldsymbol{A}+a\boldsymbol{I})^{-1}$ and pseudo Green matrix $\boldsymbol{G}_*$ are obtained and investigated in a detailed manner. The best constants of the inequalities are given by means of eigenvalues of $\boldsymbol{A}$.

In this paper, we study the fourth moment of the Epstein zeta function $\zeta (s;Q)$ associated to a $n\times n$ positive definite symmetric matrix $Q$ ($n\geq 4$) on the line $\mathrm{Re}(s)=\frac{n-1}{2}$. We prove that the integral $\int _{0}^{T}|\zeta (\frac{n-1}{2}+it;Q)|^{4}dt$ is evaluated by $O(T(\mathrm{log}\,T)^{4})$ if $Q$ satisfies some conditions. As an application, we consider the divisor problem with respect to the coefficients of the Dirichlet series of Epstein zeta functions. Certain estimates for the error term of the sum of the Dirichlet coefficients are obtained by combining our results and Fomenko's estimates for $\zeta (\frac{n-1}{2}+it;Q)$.