On the first homology of automorphism groups of manifolds with geometric structures

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category. This research was partially supported by Grant-in-Aid for Scientific Research (No. 16540058), Japan Society for the Promotion of Science. This research was partially supported by Grant-in-Aid for Scientific Research (No. 14540093), Japan Society for the Promotion of Science.     

Completely monotone sequences and universally prestarlike functions

We introduce universally convex, starlike and prestarlike functions in the slit domain ℂ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions. S. R. and L. S. received partial support from FONDECYT (grants 1070269 and 7080064) and DGIP-UTFSM (grant 240954). S. R. acknowledges partial support also from the German-Israeli Foundation (grant G-809-234.6/2003) and a JSPS Grant-in-Aid for Scientific Research (A), 17204010. T. S. was partially supported by JSPS Grant-in-Aid for Scientific Research (B), 17340039.     

Congruence of Siegel modular forms and special values of their standard zeta functions

In this paper, we consider the relationship between the congruence of cuspidal Hecke eigenforms with respect to Sp _n (Z) and the special values of their standard zeta functions. In particular, we propose a conjecture concerning the congruence between Saito-Kurokawa lifts and non-Saito-Kurokawa lifts, and prove it under certain condition. Partially supported by Grant-in-Aid for Scientific Research C-17540003, JSPS.     

Complexes of exact hermitian cubes and the Zagier conjecture

In this paper, we present a new way of constructing a map from the higher Bloch group to the higher rational K-theory for an algebraic number field. The composition of it with the regulator map is expressed in terms of the polylogarithm function. To do this, we employ exact hermitian cubes and their Bott-Chern forms.Partly supported by the JSPS Grant-in-Aid for Scientific Research (No. 14740022).     

The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

Relative mean curvature configurations for surfaces in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>, <Emphasis Type="Italic">n</Emphasis> ≥ 5

We define the relative mean curvature directions on surfaces immersed in ℝⁿ, n ≥ 4, generalizing the concept of mean curvature directions for surfaces in 4-space studied by Mello. We obtain their differential equations and study their corresponding generic configurations. *Work partially supported by DGCYT grant no. MTM2004-03244 and Unimontes-BR. †Work partially supported by DGCYT grant no. MTM2004-03244. ‡Work partially supported by DGCYT grant no. BFM2003-0203.     

Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian

We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L ²-analysis.Dedicated to Professor Hajime Urakawa on his sixtieth birthday.The author is partially supported by the Grant-in-Aid for Scientific Research No. 16740026 of the Japan Society for the Promotion of Science.     

A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions

We give a Fekete-Szeg? type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a ₂ z ² + a ₃ z ³ + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0. The first author was partially supported by the JSPS Grant-in-Aid for Scientific Research (B), 17340039. Authors’ addresses: T. Sugawa and T. Terada, Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan Current address: T. Sugawa, Division of Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan     

Jackson’s integral of the Hurwitz zeta function

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function. Partially supported by Grant-in-Aid for Exploratory Research No. 15654003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.     

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained. Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS. Hejun Sun and Hongcang Yang research were partially supported by NSF of China.     

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(=  1, 2, 3) distinct principal curvatures. Dedicated to Professor Hajime Urakawa on his sixtieth birthday. H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006.     

Lyapunov exponent for the parabolic anderson model with lévy noise

We consider the asymptotic almost sure behavior of the solution of the equationwhere {Y_x:x Z^d} is a field of independent Lévy processes and is the discrete Laplacian.Research of the first two authors supported in part by a grant from NSF, of the third author by JSPS Grant-in-Aid for Scientific Research, Kiban(C) 13640103     

Normal generation of line bundles of high degrees on smooth algebraic curves

We give necessary and sufficient conditions for non-special line bundles of degree2g — 2 and 2g — 3 being not normally generated. We also provide criteria for special line bundles of degreed > 2g — 6 being normally generated. This work was done under JSPS-KOSEF joint research program 1997. The first named author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, #10440051. The second named author was partially supported by BSRI(1998-015-D00023) and SNU(99-5-l-031). During the period when this paper was prepared for publication, the second named author was enjoying the hospitality of ICTP. The third named author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, #09640043.     

Monge’s problem with a quadratic cost by the zero-noise limit of <Emphasis Type="Italic">h</Emphasis>-path processes

We study the asymptotic behavior, in the zero-noise limit, of solutions to Schrödingers functional equations and that of h-path processes, and give a new proof of the existence of the minimizer of Monges problem with a quadratic cost.Partially supported by the Grant-in-Aid for Scientific Research, No. 15340047 and 15340051, JSPS.     

Analytic non-integrable Hamiltonian systems and irregular singularity

We study a C ^∞-Liouville-integrable and analytic non-integrable Hamiltonian system. We will show that an irregular singular character plays a crucial role in the analytic non-integrability of the system. Partially supported by Grant-in-Aid for Scientific Research (No. 11640183), Ministry of Education, Science and Culture, Japan.     

The number of Reidemeister moves for splitting a link

We give an explicit upper bound for the number of Reidemeister moves for deforming a link diagram of a split link to be disconnected.Mathematics Subject Classification (2000): 57M25, 57N10The author is partially supported by Grant-in-Aid for Scientific Research (No. 15740047), Ministry of Education, Science, Sports and Technology, Japan.     

Spectra of completely log-hyponormal operators

An operatorT on a Hilbert space is called log-hyponormal if it is invertible and log(T ^* T)≥log(T ^* T). In this paper we study spectral properties of completely log-hyponormal operators. Dedicated to professor Robin Harte on his sixtieth birthday This research is partially supported by Grant-in-Aid Scientific Research (No. 09640229).     

Anisotropic capillary surfaces with wetting energy

We study the stability of capillary surfaces for anisotropic energies having boundaries supported in horizontal planes. A wetting energy term for the surface to plane interface is included. The first author is partially supported by Grant-in-Aid for Scientific Research (C) No. 16540195 of the Japan Society for the Promotion of Science.     

On Linear Independence of Theta Values

We prove a linear independence result for the values of theta series and Tschakaloff functions T_q(z) with different values of q.The first named author was supported in part by Grant-in-Aid for Scientific Research (No. 13640007), The Ministry of Education, Science, Sports, Culture of Japan.     

The Complete Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

This is a continuation of the previous papers [3, 4, 5]. We finish determining the minimum distance of two-point codes on a Hermitian curve. Masaaki Homma: Partially supported by Grant-in-Aid for Scientific Research (15500017), JSPS. Seon Jeong Kim: Partially supported by Korea Research Foundation Grant (KRF-2004-041-C00016)