Let $f$ be a smooth function of two variables $x$, $y$ and for each positive integer $n$, let $d^{n}f$ be a symmetric tensor field of type $(0,n)$ defined by $d^{n}f:={\sum }_{i=0}^n$ $\left(\begin{array}{c}n\\ i\end{array}\right)$ $\left({\partial }_x^{n-i}{\partial }_y^{i}f\right)d{x}^{n-i}d{y}^i$ and ${\widetilde{𝒟}}_{d^{n}f}$ a finitely many-valued one-dimensional distribution obtained from $d^{n}f$: for example, ${\widetilde{𝒟}}_{d^{1}f}$ is the one-dimensional distribution defined by the gradient vector field of $f$; ${\widetilde{𝒟}}_{d^{2}f}$ consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of Hessian of $f$. In the present paper, we shall study the behavior of ${\widetilde{𝒟}}_{d^{n}f}$ around its isolated singularity in ways which appear in [1]--[4]. In particular, we shall introduce and study a conjecture which asserts that the index of an isolated singularity with respect to ${\widetilde{𝒟}}_{d^{n}f}$ is not more than one.

We describe a localization theory for Maslov classes associated with two Lagrangian subbundles in a real symplectic vector bundle and give a definition of the residue of the Maslov classes. We also compute explicitly the residue of the first Maslov class in the case that the non-transversal set of the two Lagrangian subbundles have codimension 1.

Recently, Oka-Pho proved that the fundamental group of the complement of any plane irreducible tame torus sextic is not abelian. We compute here the fundamental groups of the complements of some plane irreducible sextics which are not of torus type. For all our examples, we obtain that the fundamental group is abelian.

We prove that the asymptotic cone of every complete, connected, non-compact Riemannian manifold of roughly non-negative radial curvature exists, and it is isometric to the Euclidean cone over their Tits ideal boundaries.

In [19] a method was presented, which constructs via loop group splittings all harmonic maps into a compact symmetric space. The present paper generalizes this method to all spaces $G/K$, where $G$ is an arbitrary Lie group (semisimple or not) and $K$ is the fixpoint group of some involution of $G$. The method is illustrated by a number of examples.

If a closed oriented manifold admits an action of a finite group $G$, the equivariant determinant of a $G$-equivariant elliptic operator on the manifold defines a group homomorphism from $G$ to $S^1$. The equivariant determinant is obtained from the fixed point data of the action by using the Atiyah-Singer index theorem, and the fact that the equivariant determinant is a group homomorphism imposes conditions on the fixed point data. In this paper, using the equivariant determinant, we introduce an obstruction to the existence of a finite group action on the manifold, which is obtained directly from the relation among the generators of the finite group.

We determine an explicit form of the Igusa local zeta function of the simple prehomogeneous vector space $\left(GL\right(1{)}^4\times SL(2n+1),{\mathrm{\Lambda }}_2\oplus {\mathrm{\Lambda }}_1\oplus {\mathrm{\Lambda }}_1\oplus {\mathrm{\Lambda }}_1)$. As for the Igusa local zeta functions and the $p$-adic $\mathrm{\Gamma }$-factors of regular reducible simple prehomogeneous vector spaces with universally transitive open orbits, a table for explicit forms is completed by our result together with the result in Hosokawa [2].

We consider local properties of zero sets dependent of time variable of analytic solutions of linear second order parabolic equations with two spatial variables. Our aim is to give several necessary conditions for the zero set at the present to be homeomorphic to those at the recent past and at the recent future.

We study Mordell-Weil lattices for bielliptic fibrations on rational surfaces. We prove theorems on the structure and give an explicit construction of the fibration with maximal Mordell-Weil rank and moreover determine the structures of such lattices.

It is known that the closure of an arbitrary $K_{\mathbf{C}}$-orbit on a flag manifold is expressed as a product of a closed $K_{\mathbf{C}}$-orbit and a Schubert cell ([M2], [Sp]). We already applied this fact to the duality of orbits on flag manifolds ([GM]). We refine here this result and give its new applications to the study of domains arising from the duality.

We study dependence on a parameter of exponential attractors. As known, exponetial attractors are not uniquely determined from a dissipative dynamical system even if they exist. But we prove in this paper that one can construct an exponential attractor which depends continuously on a parameter in the dynamical system. This result is then applied to the chemotaxis-growth system.

In this paper the authors give the weak type (1,1) boundedness and the $L^p$ boundedness of a class of the parametrized Littlewood-Paley operators. These conclusions improve and complete some known results.

In this paper we characterize symmetric cones among homogeneous convex cones by the condition that the corresponding tube domains are mapped onto the dual tube domains under pseudoinverse maps with parameter. The condition also restricts the parameter to specific ones.

In this paper, we establish the formulas expressing the special values of the spectral zeta function ${\zeta }_{\mathrm{\Delta }}\left(n\right)$ of the Laplacian $\mathrm{\Delta }$ on some locally symmetric Riemannian manifold $\mathrm{\Gamma }\backslash G/K$ in terms of the coefficients of the Laurent expansion of the corresponding Selberg zeta function. As an application, we give a numerical estimation of the first eigenvalue of $\mathrm{\Delta }$ by computing the values ${\zeta }_{\mathrm{\Delta }}\left(n\right)$ numerically, when $\mathrm{\Gamma }\backslash G/K$ is a Riemann surface with $\mathrm{\Gamma }$ being the quaternion group.

In this paper we first prove that, for every hypersurface $D$ of degree $d$ in a complex projective space, there exists a holomorphic curve $f$ from the complex plane into the projective space whose deficiency for $D$ is positive and less than one. Using this result, we construct meromorphic mappings from the complex $m$-space into the complex projective space with the same properties. We also investigate the effect of resolution of singularities to defects of meromorphic mappings.

A theory of local old- and newforms for representations of $GSp\left(4\right)$ over a $p$-adic field with Iwahori-invariant vectors is developed. The results are applied to Siegel modular forms of degree 2 with square-free level with respect to various congruence subgroups.

Restricting to $K$-invariant functions we prove ${\mathbf{L}}^p$ boundedness of Fourier multipliers satisfying Hörmander-Mihlin conditions on Non-Riemannian Symmetric Spaces using Flensted-Jensen duality. We also show necessity of holomorphic extension for multipliers.