This is a continuation of our previous result, in which properties of multiple zeta-functions associated with simple Lie algebras of A_r type have been studied. In the present paper we consider more general situation, and discuss the Lie theoretic background structure of our theory. We show a recursive structure in the family of zeta-functions of sets of roots, which can be explained by the order relation among roots. We also point out that the recursive structure can be described in terms of Dynkin diagrams. Then we prove several analytic properties of zeta-functions associated with simple Lie algebras of B_r, C_r, and D_r types.

We introduce the concept of strict ample sequence in a fibered triangulated category and define the stability of the objects in a triangulated category. Then we construct the moduli space of (semi) stable objects by GIT construction.

This paper is devoted to the study of a class of semilinear degenerate elliptic boundary value problems with asymmetric nonlinearity which include as particular cases the Dirichlet and Robin problems. The most essential point is how to generalize the classical variational approach to eigenvalue problems with an indefinite weight to the degenerate case. The variational approach here is based on the theory of fractional powers of analytic semigroups. By making use of global inversion theorems with singularities between Banach spaces, we prove very exact results on the number of solutions of our problem. The results extend an earlier theorem due to Ambrosetti and Prodi to the degenerate case.

We prove that an arbitrary reducible curve on a smooth surface has an essentially unique decomposition into chain-connected curves. Using this decomposition, we give an upper bound of the geometric genus of a numerically Gorenstein surface singularity in terms of certain topological data determined by the canonical cycle. We show also that the fixed part of the canonical linear system of a 1-connected curve is always rational, that is, the first cohomology of its structure sheaf vanishes.

Intersection sheaves are usually defined for a projective flat surjective morphism of Noetherian schemes of relative dimension d and for d+1 invertible sheaves on the ambient scheme. In this article, the construction is generalized to the case of the equi-dimensional projective surjective morphisms to normal separated Noetherian schemes. Applications to the studies on family of effective algebraic cycles and on polarized endomorphisms are also given.

Given a compact Riemannian manifold M with boundary (possibly empty), we consider the eigenvalues of the biharmonic operator with weight on M, proving a general inequality involving them. Using this inequality, we consider these eigenvalues when M is a compact domain of one of the following three spaces: 1) a complex projective space, 2) a minimal submanifold of a Euclidean space and 3) a minimal submanifold of a unit sphere.

In this paper we construct nontrivial pairs of $\mathfrak{S}$-related (i.e. Smith equivalent) real G-modules for the group G = PΣL(2,27) and the small groups of order 864 and types 2666, 4666. This and a theorem of K. Pawałowski-R. Solomon together show that Laitinen's conjecture is affirmative for any finite nonsolvable gap group. That is, for a finite nonsolvable gap group G, there exists a nontrivial P(G)-matched pair consisting of $\mathfrak{S}$-related real G-modules if and only if the number of all real conjugacy classes of elements in G not of prime power order is greater than or equal to 2.

We give natural extensions for the α-Rosen continued fractions of Dajani et al. for a set of small α values by appropriately adding and deleting rectangles from the region of the natural extension for the standard Rosen fractions. It follows that the underlying maps have equal entropy.

In this paper we study eigenvalues of a clamped plate problem on a bounded domain in an n-dimensional complete Riemannian manifold. By making use of Nash's theorem and introducing k free constants, we derive a universal bound for eigenvalues, which solves a problem proposed by Wang and Xia [16].