The system of algebraic equations for the leading terms of formal solutions to the Noumi-Yamada systems with a large parameter is studied. A formula which gives the number of solutions outside of turning points is established. The number of turning points of the first kind is also given.

We establish an inequality between the dimensions of the endomorphism and extension spaces of the indecomposable modules in generalized standard almost cyclic coherent components of the Auslander-Reiten quivers of finite dimensional algebras over an arbitrary base field. As an application we provide a homological characterization, involving the Euler quadratic form, of the tame algebras with separating families of almost cyclic coherent Auslander-Reiten components.

Let M^m be an m-dimensional submanifold without umbilical points in the m+1-dimensional unit sphere S^m+1. Three basic invariants of M^m under the Möbius transformation group of S^m+1 are a 1-form Φ called Möbius form, a symmetric (0,2) tensor A called Blaschke tensor and a positive definite (0,2) tensor g called Möbius metric. We call the Blaschke tensor is isotropic if there exists a function λ such that A = λg. One of the basic questions in Möbius geometry is to classify the hypersurfaces with isotropic Blaschke tensor. When λ is constant, the classification was given by Changping Wang and others. When λ is not constant, all hypersurfaces with dimensional m ≥ 3 and isotropic Blaschke tensor are explicitly expressed in this paper. Therefore, for the dimensional m ≥ 3, the above basic question is completely answered.

This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in P³ with maximum Picard number ρ = 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before.

We study the generalized Whittaker models for G = GSp(2,R) associated with indefinite binary quadratic forms when they arise from two standard representations of G: (i) a generalized principal series representation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of G in terms of the Meijer G-functions.

We give an explicit dimension formula for the spaces of vector valued Siegel cusp forms of degree two with respect to a certain kind of arithmetic subgroups of the non-split Q-forms of Sp(2,R).

We investigate the spectral properties of the Dirac operator with a potential V(x) and two relativistic Schrödinger operators with V(x) and -V(x), respectively. The potential V(x) is assumed to be dilation analytic and diverge at infinity. Our approach is based on an abstract theorem related to dilation analytic methods, and our results on the Dirac operator are obtained by analyzing dilated relativistic Schrödinger operators. Moreover, we explain some relationships of spectra and resonances between Schrödinger operators and the Dirac operator as the nonrelativistic limit.

The main purpose of this paper is to show the nonexistence of tight Euclidean 9-designs on 2 concentric spheres in Rⁿ if n ≥ 3. This in turn implies the nonexistence of minimum cubature formulas of degree 9 (in the sense of Cools and Schmid) for any spherically symmetric integrals in Rⁿ if n ≥ 3.

Let $\Delta \subset R^n$ be an $n$-dimensional Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_\Delta$ and the very ample $(C^\times)^n$-equivariant line bundle $L_\Delta$ on $X_\Delta$ associated with $\Delta$. In the present paper, we show that if $(X_\Delta,L_\Delta^i)$ is Chow semistable then the sum of integer points in $i\Delta$ is the constant multiple of the barycenter of $\Delta$. Using this result we get a necessary condition for the polarized toric manifold $(X_\Delta,L_\Delta)$ being asymptotically Chow semistable. Moreover we can generalize the result in [4] to the case when $X_\Delta$ is not necessarily Fano.

We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.