Characterization of totally umbilic hypersurfaces in a space form by circles

In this paper we characterize totally umbilic hypersurfaces in a space form by a property of the extrinsic shape of circles on hypersurfaces. This characterization corresponds to characterizations of isoparametric hypersurfaces in a space form by properties of the extrinsic shape of geodesics due to Kimura-Maeda.The first author was partially supported by Grant-in-Aid for Scientific Research (C) No. 14540075, Ministry of Education, Science, Sports and Culture. The second author was partially supported by Grant-in-Aid for Scientific Research (C) No. 14540080, Ministry of Education, Science, Sports and Culture.     

Killing helices on a symmetric space of rank one

In this paper, we study helices which are orbits of one parameter families of isometries on a symmetric space of rank one. We introduce the notion of structure torsion fields for helices, show the necessary and sufficient condition that they are generated by some Killing vector fields, and study their moduli space. The author is partially supported by Grant-in-Aid for Scientific Research (C)(No. 14540075), Ministry of Education, Science, Sports, Culture and Technology.     

Totally Umbilic Isometric Immersions and Curves of Order Two

In this paper we study submanifolds by use of extrinsic shapes of some curves having points of proper order 2, and give a condition that they are totally umbilic. This gives an extension of Nomizu-Yano’s result in [7] on a characterization of extrinsic spheres. The second author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 17540072), Ministry of Education, Science, Sports, Culture and Technology.     

Euler factor of a certain dirichlet series attached to siegel eisenstein Series

We give an explicit form of a “good” Euler factor of a certain Dirichlet series attached to the Siegel-Eisenstein series. The author was partially supported by Grant-in-Aid for Scientific Research, the Ministry of Education, Science, Sports and Culture, Japan.     

Linear Independence of the Values of <Emphasis Type="Italic">q</Emphasis>-Hypergeometric Series and Related Functions

We give linear independence results for the values of certain entire series and of functions satisfying certain first order q-difference equations. The former generalizes a result of Bézivin, while the latter refines that of the second named author in qualitative form. These results imply linear independence of the values of q-hypergeometric series.Research supported in part by Grant-in-Aid for Scientic Research (No. 13640007), the Ministry of Education, Science, Sports and Culture of Japan.2000 Mathematics Subject Classification: Primary—11J72     

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.     

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.     

Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is two-step. New examples of Einstein solvmanifolds with three-step and four-step nilradicals are also given. This work was partially supported by Grant-in-Aid for Young Scientists (B) 14740049 and 17740039, The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Time-varying parameter auto-regressive models for autocovariance nonstationary time series

In this paper, autocovariance nonstationary time series is clearly defined on a family of time series. We propose three types of TVPAR (time-varying parameter auto-regressive) models: the full order TVPAR model, the time-unvarying order TVPAR model and the time-varying order TVPAR model for autocovariance nonstationary time series. Related minimum AIC (Akaike information criterion) estimations are carried out. This work was supported by the Doctoral Research Fund of the Ministry of Education, China (Grant No. 20040285008), and Grant-in-Aid for Scientific Research (B), the Ministry of Education, Science, Sports and Culture, Japan, 2005 (Grant No. 17300228)     

Decay and regularity for dispersive equations with constant coefficients

This article discusses some smoothing estimates of the initial value problem for dispersive equations with constant coefficients. In particular, it is shown that a certain condition for the principal part of the symbol (see the assumption (1.3) below, which is equivalent to the one “of principal type” in the paper by Ben-Artzi and Devinatz [2]) is necessary and sufficient for the maximal smoothing in space-time. Dedicated to Professor Norio Shimakura The author was supported in part by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 13640187).     

Singular points on compactifications of a Riemann surface

A notion of capacity on general compactifications of a Riemann surface is introduced. It is used to establish the connection between singular points on the Royden compactification and singular points on the Kuramochi compactification. The second named author was supported by Grant-in-Aid for Scientific Research, Japanese Ministry of Education, Science and Culture.     

A special value of the spectral zeta function of the non-commutative harmonic oscillators

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function. The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) 15340005 from the Ministry of Education, Culture, Sports, Science and Technology.     

Askey–Wilson type integrals associated with root systems

A new formula for an Askey–Wilson type integral associated with the root system F ₄ is studied. A simple proof of the evaluation formula for the original Askey–Wilson integral is also stated. 2000 Mathematics Subject Classification Primary—33D67, 33D60 This work was supported in part by Grant-in-Aid for Scientific Research (C) No. 15540045 from the Ministry of Education, Culture, Sports, Science and Technology (Japan).     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Conjugacy relationship between M-convex and L-convex functions in continuous variables

By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions in continuous variables. In this paper, we consider a further extension to more general convex functions defined over the real space, and provide a proof for the conjugacy relationship between general M-convex and L-convex functions.Mathematics Subject Classification (1991): 90C10, 90C25, 90C27, 90C35This work is supported by Grant-in-Aid of the Ministry of Education, Culture, Sports, Science and Technology of Japan     

Bifurcating critical points of bending energy under constraints related to the shape of red blood cells

Considered is a variational problem for the bending energy of closed surfaces under the prescribed area and surrounding volume. Minimizers of this problem are interpreted as surfaces modeling the shape of red blood cells. We give a rigorous proof of the existence of a one-parameter family of critical points bifurcating from the sphere and study their stability/instability. In particular, for a few branches of critical points, we compute the exact values of the index and the nullity of critical points. Received: 8 September 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 Partly supported by Grant-in-Aid for Exploratory Research (Nos.09874026, 11874033) and for Scientific Research (No.12640200), Ministry of Education, Science, Sports, and Culture, Japan; and also by Sumitomo Foundation Dedicated to Professor Takaaki Nishida on his sixtieth birthday     

Asymptotic cumulants of the parameter estimators in item response theory

The asymptotic cumulants of the parameter estimators for the three-parameter logistic model in item response theory are derived up to the fourth order with the higher-order added asymptotic variances. The asymptotic cumulants of the corresponding Studentized estimators up to the third order are also given. The estimators are obtained by marginal maximum likelihood using the standard normal distribution for the latent variable with and without model misspecification. Numerical examples with fixed guessing parameters show advantages of the asymptotic expansions over the usual normal approximation. This work was partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.     

Complete harmonic stable minimal hypersurfaces in a Riemannian manifold

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R ² or a cylinder R × S ¹, which generalizes a theorem due to Fischer-Colbrie and Schoen [12]. The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L ²-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to Miyaoka [20] and Palmer [21]. Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea. Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea     

Coordinatewise domain scaling algorithm for M-convex function minimization

We present a polynomial time scaling algorithm for the minimization of an M-convex function. M-convex functions are nonlinear discrete functions with (poly)matroid structures, which are being recognized as playing a fundamental role in tractable cases of discrete optimization. The algorithm is applicable also to a variant of quasi M-convex functions.Mathematics Subject Classification (2000):90C27, 68W40, 05B35This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.     

Adjustable Robust Optimization Models for a Nonlinear Two-Period System

We study two-period nonlinear optimization problems whose parameters are uncertain. We assume that uncertain parameters are revealed in stages and model them using the adjustable robust optimization approach. For problems with polytopic uncertainty, we show that quasiconvexity of the optimal value function of certain subproblems is sufficient for the reducibility of the resulting robust optimization problem to a single-level deterministic problem. We relate this sufficient condition to the cone-quasiconvexity of the feasible set mapping for adjustable variables and present several examples and applications satisfying these conditions. This work was partially supported by the National Science Foundation, Grants CCR-9875559 and DMS-0139911, and by Grant-in-Aid for Scientific Research from the Ministry of Education, Sports, Science and Culture of Japan, Grant 16710110.