The notion of a complex equifocal submanifold in a Riemannian symmetric space of non-compact type has been recently introduced as a generalization of isoparametric hypersurfaces in the hyperbolic space. As its subclass, the notion of a proper complex equifocal submanifold has been introduced. Some results for a proper complex equifocal submanifold have been recently obtained by investigating the lift of its complete complexification to some path space. In this paper, we give a new construction of the complete complexification of a proper complex equifocal submanifold and, by using the construction, show that leaves of focal distributions of the complete complexification are the images by the normal exponential map of principal orbits of a certain kind of pseudo-orthogonal representation on the normal space of the corresponding focal submanifold.

Three dimensional pseudo-symmetric Hopf hypersurfaces in complex projective plane and complex hyperbolic plane are classified.

Let $(X,L)$ be a hyperquadric fibration over a smooth curve with $\dim X=n\geq 3$. In this paper we will calculate the $i$th sectional Euler number $e_i(X,L)$ of $(X,L)$. Using this, we will study a lower bound for the $i$th sectional Betti number $b_i(X,L)$ with $i\leq 4$. In particular we will prove that $b_2(X,L)\geq 3$, $b_3(X,L)\geq 0$ and $b_4(X,L)\geq 3$.

We consider stability of a stationary solution of Kolmogorov flow with a bottom friction. Any solution of nonstationary problem which is periodic with respect to $x$, $y$ with periods $2\pi/\alpha$, $2\pi$ is shown to tend to the stationary solution as time tends to infinity when aspect ratio $\alpha$ is equal to or greater than 1.

In this paper we derive some subordination and superordination results for certain normalized analytic functions in the open unit disc, which are acted upon by a class of extended multiplier transformation. Relevant connection of the results, which are presented in this paper with various known results are also considered.

In this paper we study the existence of standing waves for coupled nonlinear Schrödinger equations. The interaction between equations plays an important role in our study. When the interaction is strong, the least energy solution is a solution whose both components are positive. When the interaction is weak, the least energy solution is a semitrivial solution, namely a solution of a form $(u_1,0)$ or $(0,u_2)$. Moreover, minimizing method on the Nehari type manifold with codimension 2 gives us a positive solution when the interaction is weak.

In this paper we study the Cartier duality of certain finite subgroup schemes of $\mathcal{G}^{(\lambda)}$ in positive characteristic, where $\mathcal{G}^{(\lambda)}$ denotes the form of $\mathbf{G}_m$ determined by $\lambda$. To establish the Cartier duality of these group schemes, we use certain deformations of Artin-Hasse exponential series.

For Damek-Ricci spaces $(X,g)$ we compute the exact form of the Busemann function which is needed to represent the Poisson kernel of $(X,g)$ in exponential form in terms of the Busemann function and the volume entropy. From this fact, we show that the Poisson kernel map $\varphi: (X,g) \rightarrow (\mathcal{P}(\partial X),G)$ is a homothetic embedding. Here $\mathcal{P}(\partial X)$ is the space of probability measures having positive density function on the ideal boundary $\partial X$ of $X$, and $G$ is the Fisher information metric on $\mathcal{P}(\partial X)$.

The O.D.E. studied is of second order of the Schroedinger type constaining a small parameter, considered as the Planck constant, and it has a turning point and a regular singular point both situated at the origin. A case where the turning point and the singular point coincide has a very typical figure of the existence region (canonical domain) of the asymptotic solutions. Several cases of different order of the turning point are shown.

Let $k$ be a real quadratic field. The Diophantine equation $X^3=u+27v$ in $X\in\mathcal{O}_k$ (the ring of integers of $k$), $u,v\in\mathcal{O}_k^\times$ (the group of units of $k$) is solved under some assumptions on $k$.

In this paper, we consider a quaternionic representation of a 4-dimensional Kleinian group $G$ with 3 generators $f,g$ and $h$, where $g$ and $h$ are simple parabolic, $[g,h]= id$, and $[f,g],[f,h]$ are order-2 elliptic elements. We parameterize such $f,g$ and $h$ up to conjugacy and we simulate the shape of the limit set $\Lambda(G)$ using computer.

In [2] and [3], we discuss the existence and the $(a,t)$-joint continuity of the distribution-valued additive functional $A_T(a:t,\omega)=\int^t_0 T(X_s-a)$ for $T \in H^\beta_p$ except for the case of the $(a,t)$-joint continuity with $p=1$. In this paper, we discuss the $(a,t)$-joint continuity of the distribution-valued additive functional $A_T(a:t,\omega)$ for $T \in H^\beta_1$.

In this paper, we shall consider an initial value problem of the differential equation written in the title where \[ '=d/dt ,\quad -(2\lambda+1)^2/4\lambda(\lambda+1) < \alpha < 0 ,\quad \lambda < -1. \] For every initial condition, we shall obtain analytical expressions of the solution of the initial value problem. These analytical expressions are valid in neighborhoods of both ends of the domain of the solution, and hence show asymptotic behavior of the solution.

We say that a regular graph $G$ of order $n$ and degree $r\ge 1$ (which is not the complete graph) is strongly regular if there exist non-negative integers $\tau$ and $\theta$ such that $|S_i\cap S_j| = \tau$ for any two adjacent vertices $i$ and $j$, and $|S_i\cap S_j| = \theta$ for any two distinct non-adjacent vertices $i$ and $j$, where $S_k$ denotes the neighborhood of the vertex $k$. We say that a graph $G$ of order $n$ is walk regular if and only if its vertex deleted subgraphs $G_i = G\smallsetminus i$ are cospectral for $i = 1,2,\ldots ,n$. We here establish necessary and sufficient conditions under which a walk regular graph $G$ which is cospectral to its complement $\overline G$ is strongly regular.

We present elements of a theory of translation-invariant integration, measure, and harmonic analysis on a valuation field with local field as residue field, extending work of I. Fesenko. Applications to zeta integrals for two-dimensional local fields are then considered.